Урок 1, 2
|Óðîê 1 – 3 |2.09. |
|Òðèãîíîìåòðè÷åñêèå ôóíêöèè ÷èñëîâîãî àðãóìåíòà. |
|×åòíîñòü, íå÷åòíîñòü è ïåðèîäè÷íîñòü òðèãîíîìåòðè÷åñêèõ ôóíêöèé. |
1. Ââåäåíèå.
À) Ðàññàäèòü øêîëüíèêîâ. Íàïîìíèòü î äèñöèïëèíå: ïðèõîä â êëàññ, ïîäãîòîâêà ê óðîêó, ïîäãîòîâêà äîñêè.
Á) Ìàòåìàòèêà – 9 ÷àñîâ â íåäåëþ.
Ïî îäíîé òîëñòîé ðàáî÷åé òåòðàäè äëÿ êàæäîãî ïðåäìåòà; îäíà òîíêàÿ – äëÿ ê/ð; îòäåëüíàÿ òåòðàäü äëÿ í/îá çàäàíèé; îäèíàðíûå ëèñòî÷êè äëÿ ñ/ð. Ëèíåéêà, óãîëüíèê, öèðêóëü – íà êàæäûé óðîê. Öâåòíûå ðó÷êè èëè ôëîìàñòåðû!
Â). Ñèñòåìà îöåíîê ïðåæíÿÿ: «áàëëüíûå» ê/ð; ñ/ð; çà÷åòû (ïî àëãåáðå – 3; ïî ãåîìåòðèè – 2); ýêçàìåí ïî àëãåáðå è ìàòåìàòè÷åñêîìó àíàëèçó.
Ã) Ïî àëãåáðå è ìàòåìàòè÷åñêîìó àíàëèçó – ó÷åáíèê Í.ß. Âèëåíêèíà; ïî ãåîìåòðèè – ó÷åáíèê À.Ä. Àëåêñàíäðîâà (ðàçäàòü). Ê çàâòðàøíåìó óðîêó ãåîìåòðèè ïðî÷èòàéòå ââåäåíèå (ñòð. 3 – 7).
2. Íîâûé ìàòåðèàë.  êóðñå àëãåáðû ìû ïðîäîëæàåì èçó÷àòü òðèãîíîìåòðèþ, íî åñëè â 9 êëàññå îñíîâíîé óïîð äåëàëñÿ íà çíàíèå ôîðìóë è òðèãîíîìåòðè÷åñêèå ïðåîáðàçîâàíèÿ, òî â 10 êëàññå ìû çàéìåìñÿ òðèãîíîìåòðè÷åñêèìè ôóíêöèÿìè.
Âñïîìíèòå: 1) îïðåäåëåíèå ôóíêöèè; 2) ÷òî òàêîå îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé ôóíêöèè; 3) ÷òî òàêîå ÷èñëîâàÿ ôóíêöèÿ; 4) îïðåäåëåíèÿ ÷åòíîé è íå÷åòíîé ôóíêöèé.
[1) Ôóíêöèåé íàçûâàåòñÿ ñîîòâåòñòâèå ìåæäó ìíîæåñòâàìè À è Â, ïðè êîòîðîì êàæäîìó ýëåìåíòó ìíîæåñòâà À ñîîòâåòñòâóåò íå áîëåå îäíîãî ýëåìåíòà ìíîæåñòâà Â. 2) Îáëàñòü îïðåäåëåíèÿ ôóíêöèè – ìíîæåñòâî ïðîîáðàçîâ, ìíîæåñòâî çíà÷åíèé ôóíêöèè – ìíîæåñòâî îáðàçîâ. 3) Îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé – ÷èñëîâûå ìíîæåñòâà (ïîäìíîæåñòâà R). 4) Ôóíêöèÿ y = f(x) íàçûâàåòñÿ ÷åòíîé, åñëè åå îáëàñòü îïðåäåëåíèÿ ñèììåòðè÷íà îòíîñèòåëüíî íóëÿ è f(–x) = f(x). Ôóíêöèÿ y = f(x) íàçûâàåòñÿ íå÷åòíîé, åñëè åå îáëàñòü îïðåäåëåíèÿ ñèììåòðè÷íà îòíîñèòåëüíî íóëÿ è f(–x) = –f(x)]
[pic]
Ðàññìîòðèì åäèíè÷íóþ îêðóæíîñòü (ñì. ðèñ. 1) è âñïîìíèì îïðåäåëåíèÿ òðèãîíîìåòðè÷åñêèõ ôóíêöèé ïðîèçâîëüíûõ óãëîâ.
Ðèñ. 1
Ïóñòü [pic], ãäå (((–(; +() è Ð((x(; y().
Òîãäà cos( = x(; sin( = y(; tg( = [pic]; ctg( = [pic] (ïðîãîâîðèòü).
Îïðåäåëåíèå. Òðèãîíîìåòðè÷åñêîé ôóíêöèåé ÷èñëà ( íàçûâàåòñÿ ñîîòâåòñòâóþùàÿ òðèãîíîìåòðè÷åñêàÿ ôóíêöèÿ óãëà â ( ðàäèàí.
Êàê ýòî ïîíèìàòü?
Ðàññìîòðèì ñîîòâåòñòâèå f: R ( îêð (0; 1) | (x(R f(x) = Px = [pic]. Íàãëÿäíî ýòî ñîîòâåòñòâèå ïðåäñòàâëÿåòñÿ íàìàòûâàíèåì áåñêîíå÷íîé íèòè (÷èñëîâîé îñè) íà áàðàáàí (åäèíè÷íóþ îêðóæíîñòü), ïðè÷åì ÷èñëî 0 çàêðåïëÿåòñÿ â òî÷êå Ð0 (ïîêàçàòü íà ðèñóíêå).
1) Îáúÿñíèòå, ïî÷åìó ýòî ñîîòâåòñòâèå ÿâëÿåòñÿ ôóíêöèåé; 2) óêàæèòå åå îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé; 3) âåðíî ëè, ÷òî ðàçëè÷íûì çíà÷åíèÿì x ñîîòâåòñòâóþò ðàçëè÷íûå çíà÷åíèÿ ôóíêöèè? ×òî èç ýòîãî ñëåäóåò?
Òàêèì îáðàçîì, òåïåðü ìû ìîæåì ðàññìàòðèâàòü ÷èñëîâûå ôóíêöèè, óêàçàâ (íà îñíîâàíèè îïðåäåëåíèé, èñïîëüçóÿ åäèíè÷íóþ îêðóæíîñòü) èõ îáëàñòè îïðåäåëåíèÿ è ìíîæåñòâà çíà÷åíèé.
|y = cos( |y = sin( |y = tg( |y = ctg( |
|D(y) = R |D(y) = R |D(y) = {((R | ( ( 0,5( + (n, n(Z} |D(y) = {((R | ( ( (n, n(Z} |
|E(y) = [–1; 1] |E(y) = [–1; 1] |E(y) = R |E(y) = R |
[pic]
Ðèñ. 2
Âñïîìíèòå, êàêèå èç òðèãîíîìåòðè÷åñêèõ ôóíêöèé ÿâëÿþòñÿ ÷åòíûìè, à êàêèå – íå÷åòíûìè. Êàê ýòî äîêàçûâàòü? (Ïðîãîâîðèòü ïî ðèñ. 2 ñ êðàòêèìè çàïèñÿìè íà äîñêå.)
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå; çàïèñè: äëÿ ïåðâîãî ïóíêòà îïðåäåëåíèÿ – äâå âîçìîæíûå ôîðìû!):
1) Äîêàæèòå, ÷òî ôóíêöèÿ à) [pic] ÿâëÿåòñÿ ÷åòíîé; á) [pic] ÿâëÿåòñÿ íå÷åòíîé. 2) Èññëåäóéòå íà ÷åòíîñòü è íå÷åòíîñòü ôóíêöèþ [pic] [D(f) = {x(R | x ( (1}; Í]
4. Óñòíî: 1) Îïðåäåëèòå ÷åòíîñòü ëèáî íå÷åòíîñòü ôóíêöèè è êðàòêî îáîñíóéòå:
|F |× |Í |× |Í |
|G |× |Í |Í |× |
|kF ( mG, k(R, |× |Í |– |– |
|k ( 0; m(R, m ( 0 | | | | |
|F(G |× |× |Í |Í |
|[pic] |× |× |Í |Í |
|F(G) |× |Í |× |× |
à) |tgx| [×]; á) sin|(| [×]; â) [pic] [Í]; ã) sin((cos((tg((ctg( + 2 [–]; ä) [pic] [×]; å) [pic] [Í]; æ) [pic] [–]; ç) ctg(t0,2) [–]; è) [pic] [Í]; ê) sin(cos(tg(ctgx))) [×]; ë) [sinx] [–]; ì) cos[x] [–]; í) {cosx} [×]; î) sin{x} [–].
2) Çàïîëíèòå òàáëèöó íà äîñêå è îáîñíóéòå:
5. Íîâûé ìàòåðèàë. Ðàññìîòðèì åùå îäíî îáùåå ñâîéñòâî ôóíêöèé.
Îïðåäåëåíèå. Ôóíêöèÿ f(x) íàçûâàåòñÿ ïåðèîäè÷åñêîé, åñëè ( T ( 0 | (x(D(f) âåðíî, ÷òî è x ( T(D(f) è âûïîëíÿåòñÿ ðàâåíñòâî f(x + T) = f(x).  ýòîì ñëó÷àå ÷èñëî Ò íàçûâàåòñÿ ïåðèîäîì ýòîé ôóíêöèè.
Ïðèìåðû. 1) {x} (T = 1); 2) ëþáàÿ èç îñíîâíûõ òðèãîíîìåòðè÷åñêèõ ôóíêöèé (T = 2().
Âîïðîñû. 1) Ïî÷åìó â ïåðâîé ÷àñòè îïðåäåëåíèÿ çíàê «(», à âî âòîðîé – òîëüêî «+»?
2) ßâëÿåòñÿ ëè ÷èñëî Ò åäèíñòâåííûì?
Îïðåäåëåíèå. Îñíîâíûì ïåðèîäîì ïåðèîäè÷åñêîé ôóíêöèè íàçûâàåòñÿ åå íàèìåíüøèé ïîëîæèòåëüíûé ïåðèîä.
Òåîðåìà. Îñíîâíûì ïåðèîäîì ôóíêöèé ñèíóñ è êîñèíóñ ÿâëÿåòñÿ T = 2(, à ôóíêöèé òàíãåíñ è êîòàíãåíñ – T = (.
Äîêàçàòåëüñòâî. I. f(x) = sinx è f(x) = cosx.
À) 1) D(f) = R ((x(D(f) âåðíî, ÷òî è x ( 2((D(f). 2) (x(D(f) f(x + 2() = f(x), òàê êàê sin(x + 2() = sinx è cos(x + 2() = cosx. Ñëåäîâàòåëüíî, 2( – ïåðèîä êàæäîé èç ôóíêöèé.
Á) Ïóñòü (Ò | 0 < T < 2( è Ò – ïåðèîä ôóíêöèè. Òîãäà: 1) (x(R sin(x + T) = sinx.
Ïðè [pic] ïîëó÷èì, ÷òî [pic] ( cosT = 1 ( T = 2(n, ãäå n(Z, ÷òî ïðîòèâîðå÷èò âûáîðó ÷èñëà Ò. 2) (x(R cos(x + T) = cosx. Ïðè [pic] ïîëó÷èì, ÷òî cosT = 1 ( T = 2(n, ãäå n(Z, ÷òî ïðîòèâîðå÷èò âûáîðó ÷èñëà Ò.
Òàêèì îáðàçîì, 2( – îñíîâíîé ïåðèîä, ÷òî è òðåáîâàëîñü äîêàçàòü.
II. g(x) = tgx è h(x) = ctgx.
À) 1) D(g) = {x(R | x ( 0,5( + (n, n(Z}; D(h) = {x(R | x ( (n, n(Z} ((x(D âåðíî, ÷òî è x ( ((D (åäèíè÷íàÿ îêðóæíîñòü!). 2) (x(D tg(x + () = tgx è ñtg(x + () = ñtgx (ôîðìóëû ïðèâåäåíèÿ), ïîýòîìó, ( – ïåðèîä êàæäîé èç ôóíêöèé.
Á) Ïóñòü (Ò | 0 < T < ( è Ò – ïåðèîä ôóíêöèè. Òîãäà: 1) (x(D tg(x + T) = tgx. Ïðè [pic] ïîëó÷èì, ÷òî tgT = 0 ( T = (k, ãäå k(Z, ÷òî ïðîòèâîðå÷èò âûáîðó ÷èñëà Ò.
2) (x(D ñtg(x + T) = ñtgx. Ïðè [pic] ïîëó÷èì, ÷òî [pic] ( –tgT = 0 ( T = (k, ãäå k(Z, ÷òî ïðîòèâîðå÷èò âûáîðó ÷èñëà Ò.
Òàêèì îáðàçîì, ( – îñíîâíîé ïåðèîä, ÷òî è òðåáîâàëîñü äîêàçàòü.
6. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå; çàïèñè!):
Äîêàæèòå, ÷òî 3( ÿâëÿåòñÿ ïåðèîäîì ôóíêöèè y = tg[pic]x – ctg2x [D(y) = {x(R | x ( [pic], n(Z}].
Äîìàøíåå çàäàíèå: ïîâòîðèòå òðèãîíîìåòðè÷åñêèå ôîðìóëû, × è Í òðèãîíîìåòðè÷åñêèõ ôóíêöèé (åñòü â ó÷åáíèêå); îïðåäåëåíèÿ è òåîðåìà î ïåðèîäè÷íîñòè – ïî òåòðàäè; 1) Èññëåäóéòå íà × è Í ôóíêöèè: à) f(x) = cos4x + 2sin1,5x(sinx + 5x2; á) [pic]; â) [pic]; ã) g(x) = (sinx)0,5; ä) h(x) = [pic]. 2) Äîêàæèòå, ÷òî 1 – îñíîâíîé ïåðèîä ôóíêöèè {x}. 3) Äîêàæèòå, ÷òî f(x) = 4ctg3x + 5sin4x ïåðèîäè÷íà ñ ïåðèîäîì (. 4) Äîêàæèòå, (f(x) (g(x) è h(x) | f(x) = g(x)sinx + h(x)cosx.
|Óðîê 4, 5 |4.09. |
|Îáùèå ñâîéñòâà ïåðèîäè÷åñêèõ ôóíêöèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû? 1) – îòâåòû? [×; Í; ×; –; –]. Êàê íàõîäèëè îáëàñòü îïðåäåëåíèÿ â ïóíêòå â)? [sint < t (åäèíè÷íàÿ îêðóæíîñòü)!]; 2) Â.: ñòð. 233 – 234; 3) [D(f) = {x(R | x ( [pic], n(Z}]; 4) [Ðàññìîòðèì, íàïðèìåð, ôóíêöèè g(x) = f(x)(sinx è h(x) = f(x)(cosx. Òîãäà, g(x)(sinx + h(x)(cosx = f(x)(sin2x + cos2x) = f(x), ÷òî è òðåáîâàëîñü äîêàçàòü]
2. Óñòíî: ïðèâåäèòå ïðèìåð ïåðèîäè÷åñêîé ôóíêöèè, ïåðèîäîì êîòîðîé: à) ÿâëÿåòñÿ ëþáîå äåéñòâèòåëüíîå ÷èñëî; á) ÿâëÿåòñÿ ëþáîå ðàöèîíàëüíîå ÷èñëî, íî íå ÿâëÿåòñÿ íèêàêîå èððàöèîíàëüíîå [à) f(x) = c, ñ(R; á) [pic] – ôóíêöèÿ Äèðèõëå]
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì îáùèå ñâîéñòâà ïåðèîäè÷åñêèõ ôóíêöèé.
1) Åñëè ôóíêöèè f(x) è g(x) èìåþò ïåðèîä Ò, òî è ôóíêöèè h(x) = kf(x) ( mg(x), ãäå k(R è m(R; p(x) = f(x)(g(x); [pic] èìåþò òîò æå ïåðèîä.
Äîêàçàòåëüñòâî. À) D(h) = D(p) = D(f)(D(g); D(q) = {x( D(f)(D(g) | g(x) ( 0}. Òàê êàê f(x) è g(x) èìåþò ïåðèîä Ò, òî x ( T(D(f) è x ( T(D(g), ñëåäîâàòåëüíî, x ( T(D(f)(D(g). Ðàññìîòðèì x | g(x) ( 0, òîãäà g(x + T) = g(x) ( 0, òî åñòü, (x(D âåðíî, ÷òî è x ( Ò(D.
Á) (x(D h(x + Ò) = kf(x + Ò) ( mg(x + Ò) = kf(x) ( mg(x) = h(x); p(x + T) = f(x + T)(g(x + T) = f(x)(g(x) = p(x); [pic].
Òàêèì îáðàçîì, óòâåðæäåíèå äîêàçàíî.
Ïðèìåðû (óêàæèòå ïåðèîä ôóíêöèè, âîñïîëüçîâàâøèñü äîêàçàííîé òåîðåìîé, è íàéäèòå äðóãîé ñïîñîá îáîñíîâàíèÿ). À) h(x) = [pic]sinx + cosx; T = 2( [h(x) = 2sin(x + [pic])] Á) p(x) = sinx(cosx; T = 2( [p(x) = 0,5sin2x] Â) [pic]; T = 2( [q(x) = tgx]
 äîêàçàííîì óòâåðæäåíèè íå ãîâîðèòñÿ î òîì, ÷òî íàéäåííûé ïåðèîä – îñíîâíîé, ÷òî âèäíî èç ïðèìåðîâ á) è â)!
2) Åñëè Ò – ïåðèîä ôóíêöèè f(x), òî è –Ò ÿâëÿåòñÿ ïåðèîäîì f(x).
Äîêàçàòåëüñòâî. À) (x(D(f) âåðíî, ÷òî è x ( (–T) = x [pic] T(D(f).
Á) (x’(D(f) f(x + (–T)) = f(x – T) = f(x), òàê êàê (x’(D(f) f(x’ + T) = f(x’) (x’ = x – T).
3) Åñëè T1 è Ò2 – ïåðèîäû ôóíêöèè f(x), òî è T = Ò1 + Ò2 ÿâëÿåòñÿ ïåðèîäîì f(x).
Äîêàçàòåëüñòâî. À) (x(D(f) âåðíî, ÷òî è x ( T1(D(f), ñëåäîâàòåëüíî (x ( Ò1) ( Ò2 = x ( (Ò1 + Ò2) = x ( T (D(f).
Á) (x(D(f) f(x + T) = f(x + (Ò1 + Ò2)) = f((x + Ò1) + Ò2) = f(x + Ò1) = f(x).
Ñëåäñòâèå. (n(N åñëè Ò – ïåðèîä ôóíêöèè f(x), òî è nÒ ÿâëÿåòñÿ ïåðèîäîì f(x).
Äîêàçàòåëüñòâî. Äëÿ n = 1 – î÷åâèäíî.
Ïóñòü óòâåðæäåíèå âåðíî ïðè n = k, äîêàæåì, ÷òî îíî âåðíî ïðè n = k + 1.
À) (x(D(f) âåðíî, ÷òî è x ( kT(D(f), ñëåäîâàòåëüíî, x ( (k + 1)T = (x ( kT) ( T(D(f).
Á) (x(D(f) f(x + (k + 1)T) = f((x + kT) + T) = f(x + kT) = f(x).
Ïî÷åìó ýòî ñâîéñòâî âåðíî (n(Z \ {0}? [ñâîéñòâî 2]
4) Åñëè T – îñíîâíîé ïåðèîä ôóíêöèè f(x), òî {nT | n(Z \ {0}} – ìíîæåñòâî âñåõ ïåðèîäîâ f(x).
Äîêàçàòåëüñòâî. Òî, ÷òî ýòè ÷èñëà ÿâëÿþòñÿ ïåðèîäàìè – óæå äîêàçàíî.
Ïóñòü (Ò’ > 0 – ïåðèîä ôóíêöèè è Ò’({nT | n(Z}. Òàê êàê T – îñíîâíîé ïåðèîä, òî Ò’ > T, òî åñòü, (k(N | kT < T’ < (k + 1)T. Òîãäà T’’ = T’ – kT – òàêæå ïåðèîä ôóíêöèè (ñâîéñòâà 2 è 3), ïðè÷åì, 0 < T’’ < T, ÷òî ïðîòèâîðå÷èò óñëîâèþ. Ñëó÷àé, êîãäà T’ < 0 ñâîäèòñÿ ê ðàññìîòðåííîìó ïî ñâîéñòâó 2.
Ñâîéñòâà 2 – 4 âìåñòå ñ ðàíåå äîêàçàííûìè íàìè óòâåðæäåíèÿìè ïîçâîëÿþò óêàçàòü ìíîæåñòâî ïåðèîäîâ ýëåìåíòàðíûõ ôóíêöèé: äëÿ {x} –Z \ {0}; äëÿ sinx è cosx – {2(n}; äëÿ tgx è ctgx – {(n}, ãäå n(Z \ {0}.
5) Ãðàôèê ôóíêöèè f(x), èìåþùåé ïåðèîä Ò, îòîáðàæàåòñÿ íà ñåáÿ ïðè [pic], ãäå [pic], (n(Z \ {0}.
Äëÿ ïîñòðîåíèÿ êàêîãî ãðàôèêà ýòî ñâîéñòâî óæå ïðèìåíÿëîñü? [{x}]
Äîêàçàòåëüñòâî. (n(Z \ {0} f(x + nT) = f(x), ÷òî è îïðåäåëÿåò ñîîòâåòñòâóþùèé ïàðàëëåëüíûé ïåðåíîñ âäîëü îñè x.
Ñâîéñòâà 6) è 7) ìû ðàññìîòðèì íà ñëåäóþùåì óðîêå (ìîæíî îñòàâèòü ìåñòî).
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå; çàïèñè!):
1) Âû÷èñëèòå: [pic] [[pic]].
2) Ïðèâåäèòå ê çíà÷åíèþ òðèãîíîìåòðè÷åñêîé ôóíêöèè íàèìåíüøåãî ïîëîæèòåëüíîãî àðãóìåíòà: à) ctg673( [–tg43(]; á) [pic] [[pic]]; â) sin(2 – 7,2() [[pic]]
3) Äîêàæèòå, ÷òî ( íå ÿâëÿåòñÿ ïåðèîäîì ôóíêöèè y = tg(sinx).
[Îò ïðîòèâíîãî: (x(D(y) –tg(sinx) = tg(sinx), ÷òî íå âûïîëíÿåòñÿ, íàïðèìåð, ïðè [pic]]
4) ßâëÿåòñÿ ëè ïåðèîäè÷åñêîé ôóíêöèÿ h(x) = [pic]?
[Íåò, òàê êàê D(h) = [0; +(), òî åñòü, (x(D(h) | x – T(D(h)]
5) Äîêàæèòå, ÷òî ôóíêöèÿ g(x) = {x2} íå ÿâëÿåòñÿ ïåðèîäè÷åñêîé.
[Îò ïðîòèâíîãî, òîãäà (x(R {(x + T)2} = {x2}. Ïðè x = 0 ïîëó÷èì, ÷òî Ò2(Z; òîãäà ïðè x = 1 ïîëó÷èì, ÷òî Ò(Q, à ïðè x = [pic] ïîëó÷èì, ÷òî Ò(Q – ïðîòèâîðå÷èå!]
Ñëåäóþùèé óðîê – ñ/ð!
Äîìàøíåå çàäàíèå: òåîðèÿ – ïî òåòðàäè è Â.: ñòð. 232 – 234; ïîâòîðèòå ôîðìóëû ïðèâåäåíèÿ è îñíîâíûå ñâîéñòâà òðèãîíîìåòðè÷åñêèõ ôóíêöèé. Äîêàæèòå, ÷òî: à) 15 ÿâëÿåòñÿ ïåðèîäîì ôóíêöèè y = {2,6x}; á) [pic] ÿâëÿåòñÿ ïåðèîäîì ôóíêöèè f(x) = ctg(2x – 1); â) ôóíêöèÿ g(x) = [pic] ÿâëÿåòñÿ ÷åòíîé è ÷èñëî ( íå ÿâëÿåòñÿ åå ïåðèîäîì; ã) [pic] íå ÿâëÿåòñÿ ïåðèîäîì ôóíêöèè y = |sinx| – cosx. Â.: ¹518 (1); ¹506 (1).
|Óðîê 6, 7 |9.09. |
|Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹1. Ñâîéñòâà è ãðàôèêè òðèãîíîìåòðè÷åñêèõ ôóíêöèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû? [â) ïðîòèâîðå÷èå ïðè x = 1; ã) ïðîòèâîðå÷èå ïðè x = 0].
¹518 (1) [îò ïðîòèâíîãî, ìîæíî âçÿòü, íàïðèìåð, x = 0 è x = 4(2]
2. Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹1 (íà ëèñòî÷êàõ; 15 – 20 ìèíóò).
3. Íîâûé ìàòåðèàë. Íà îñíîâàíèè èçâåñòíûõ íàì ñâîéñòâ òðèãîíîìåòðè÷åñêèõ ôóíêöèé ìîæíî ïîñòðîèòü èõ ãðàôèêè.
Âûïèøåì ñâîéñòâà ôóíêöèè y = sinx, àíàëèçèðóÿ êàê îòðàçèòñÿ íà ãðàôèêå êàæäîå èç íèõ. Èçîáðàçèì äåêàðòîâó ñèñòåìó êîîðäèíàò (íà äîñêå è â òåòðàäÿõ).
|1) D(y) = R |Ñóùåñòâóþò òî÷êè ñ ëþáûìè àáñöèññàìè |
|2) E(y) = [–1; 1] |Ãðàôèê ëåæèò â ïîëîñå ìåæäó ïðÿìûìè y = –1 è y = 1 |
|3) 2( – îñíîâíîé ïåðèîä |Èññëåäóåì íà [–(;(] |
|4) ôóíêöèÿ – íå÷åòíàÿ |Èññëåäóåì íà [0;(] |
|5) à) sin(( – x) = sinx |Èññëåäóåì íà [0;0,5(] |
|á) sin0 = 0; sin[pic] = 1; sin[pic] = 0,5; sin[pic] ( 0,7; |Îòìå÷àåì ñîîòâåòñòâóþùèå òî÷êè â ñèñòåìå êîîðäèíàò, çàòåì ñòðîèì ãðàôèê íà [0;0,5(]. Èñïîëüçóÿ|
|sin[pic] ( 0,85; |óêàçàííûå ñâîéñòâà ôóíêöèè ïîëó÷àåì ãðàôèê íà R. |
| |Îí íàçûâàåòñÿ ñèíóñîèäîé |
Îñòàëüíûå ñâîéñòâà ôóíêöèè ìîæíî ïîëó÷èòü, èñïîëüçóÿ ïîñòðîåííûé ãðàôèê:
|6) Íóëè ôóíêöèè: y = 0 ïðè x = (n, n(Z |
|7) Ïðîìåæóòêè çíàêîïîñòîÿíñòâà: y > 0 ïðè x([pic]((n; ( + (n); y < 0 ïðè x([pic]( –( + (n; (n) |
|8) Ìîíîòîííîñòü: ôóíêöèÿ âîçðàñòàåò íà êàæäîì ïðîìåæóòêå âèäà [pic]; óáûâàåò íà êàæäîì ïðîìåæóòêå âèäà [pic]; ãäå n(Z |
|9) Ýêñòðåìàëüíûå çíà÷åíèÿ ôóíêöèè: |
|max(sinx) = 1 ïðè x = [pic]; min(sinx) = –1 ïðè x = [pic]; ãäå n(Z |
4. Óïðàæíåíèÿ. Ïîëüçóÿñü ïîñòðîåííûì ãðàôèêîì:
1) Âû÷èñëèòå: à) sin[pic]; á) sin[pic] [à) (–0,3; á) –0,5]
2) Íàéäèòå x | à) sinx = 0,5; á) sinx = [pic] [à) x = [pic] èëè x = [pic], n(Z; á) x = [pic] èëè x = [pic], n(Z]
3) Îïðåäåëèòå çíàêè ÷èñåë: à) sin1,9(; á) sin(–3,5) [à) –; á) +]
4) Ñðàâíèòå: à) sin[pic] è sin[pic]; á) sin(–5) è sin(–6) [a) >; á) >]
5. Íîâûé ìàòåðèàë. Âûïèøåì ñâîéñòâà ôóíêöèè y = tgx, àíàëèçèðóÿ êàê îòðàçèòñÿ íà ãðàôèêå êàæäîå èç íèõ. Èçîáðàçèì äåêàðòîâó ñèñòåìó êîîðäèíàò (íà äîñêå è â òåòðàäÿõ).
|1) D(y) = {x(R | x ( 0,5( + (n, n(Z} |Íå ñóùåñòâóþò òî÷åê ñ òàêèìè àáñöèññàìè |
|2) E(y) = R |Ñóùåñòâóþò òî÷êè ñ ëþáûìè îðäèíàòàìè |
|3) ( – îñíîâíîé ïåðèîä |Èññëåäóåì íà (–0,5(; 0,5() |
|4) ôóíêöèÿ – íå÷åòíàÿ |Èññëåäóåì íà [0; [pic]) |
|5) tg0 = 0; tg[pic] = [pic] ( 0,55; |Îòìå÷àåì ñîîòâåòñòâóþùèå òî÷êè â ñèñòåìå êîîðäèíàò, çàòåì ñòðîèì ãðàôèê íà [0; [pic]). |
|tg[pic] = 1; tg[pic] ( 1,7 |Èñïîëüçóÿ óêàçàííûå ñâîéñòâà ôóíêöèè ïîëó÷àåì ãðàôèê íà D. |
| |Îí íàçûâàåòñÿ òàíãåíñîèäîé (àñèìïòîòû!) |
Îñòàëüíûå ñâîéñòâà ôóíêöèè ìîæíî ïîëó÷èòü, èñïîëüçóÿ ïîñòðîåííûé ãðàôèê:
|6) y = 0 ïðè x = (n, n(Z |
|7) y > 0 ïðè x([pic]((n; 0,5( + (n); y < 0 ïðè x([pic]( –0,5( + (n; (n) |
|8) Ôóíêöèÿ âîçðàñòàåò íà êàæäîì ïðîìåæóòêå âèäà [pic], ãäå n(Z; ïðîìåæóòêîâ óáûâàíèÿ íåò |
|9) Ýêñòðåìàëüíûõ çíà÷åíèé ôóíêöèè íå ñóùåñòâóåò |
|10) Àñèìïòîòû: ïðÿìûå, èìåþùèå óðàâíåíèÿ [pic] |
6. Óïðàæíåíèÿ. Ïîëüçóÿñü ïîñòðîåííûì ãðàôèêîì:
1) Âû÷èñëèòå: à) tg[pic]; á) tg[pic] [à) (5; á) 1]
2) Íàéäèòå x | à) tgx = [pic]; á) tgx = [pic] [à) x = [pic], n(Z; á) x = [pic], n(Z]
3) Îïðåäåëèòå çíàêè ÷èñåë: à) tg([pic]; á) tg(–4) [à) +; á) –]
4) Ñðàâíèòå: à) tg[pic] è tg[pic]; á) tg(–1,5) è tg(–1) [a) 0, l > 0, òî ÍÎÊ (k; l) = m(R | [pic]; [pic] è m – íàèìåíüøåå.
Ïðèìåðû è óïðàæíåíèÿ (1) – 4) – ïîäáîð!).
1) ÍÎÊ (5,2; 1) = 26; 2) ÍÎÊ [pic] = 1; 3) ÍÎÊ (1,4; 2,1) = 4,2; 4) ÍÎÊ [pic] = 12[pic]; 5) ÍÎÊ [pic] = ÍÎÊ [pic] = 13, òàê êàê ÍÎÊ (5; 7) = 1; 6) ÍÎÊ [pic] = ÍÎÊ [pic] = 10; 7) ÍÎÊ [pic] íå ñóùåñòâóåò, òàê êàê: ïóñòü [pic] è [pic], òîãäà 2n[pic] = k[pic] ( [pic] – ïðîòèâîðå÷èå.
Òåïåðü ìû ãîòîâû ðàññìîòðåòü ïîñëåäíåå ñâîéñòâî ïåðèîäè÷åñêèõ ôóíêöèé.
7) Åñëè Ò – îñíîâíîé ïåðèîä ôóíêöèé f(x) è g(x), òî îñíîâíûì ïåðèîäîì ôóíêöèè h(x) = f(px) + g(qx) ÿâëÿåòñÿ T’ = mT, ãäå m = ÍÎÊ [pic].
Äîêàçàòåëüñòâî – í/îá íà äîì.
Ïðèìåð. Íàéäåì îñíîâíîé ïåðèîä ôóíêöèè f(x) = 3sin5,3x – cos11x.
Îñíîâíîé ïåðèîä ôóíêöèé sinx è cosx ðàâåí 2(; p = 5,3; q = 11; m = ÍÎÊ [pic] = ÍÎÊ [pic] = 10; T’ = 20(.
6. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé):
1) Â.: ñòð. 234, ¹508 (1, 2, 5) [1) 15; 2) 1; 5) 10];
2) Â.: ñòð. 237, ¹517 (2, 4) [2) 80(; 4) ( (÷àñòíîå ôóíêöèé ñ îäèíàêîâûì ïåðèîäîì!)]
3) h(x) = 4ctg3x + 5sin4x [(; ( – îñíîâíîé ïåðèîä ôóíêöèé ctgx è sin2x; ÍÎÊ [pic] = 1]
4) Â.: ñòð. 238, ¹518 (3)
[Îò ïðîòèâíîãî; ÍÎÊ [pic] íå ñóùåñòâóåò: åñëè m(N, òî [pic]]
Äîìàøíåå çàäàíèå: òåîðèÿ – ïî òåòðàäè; äîêàçàòåëüñòâî ñâîéñòâà 7; ïîâòîðèòå ïðåîáðàçîâàíèÿ ãðàôèêîâ íà êîîðäèíàòíîé ïëîñêîñòè. Â.: ¹508 (4, 7); ¹517 (1, 5). Íàéäèòå îñíîâíûå ïåðèîäû ôóíêöèé (à) – â) – ïîñòðîéòå ãðàôèêè): à) {2x}; á) 0,5sin0,5x; â) 3ctg0,2x; ã) [pic]; ä) –cos1,6x; å) 0,1tg(0,1(x).
|Óðîê 10, 11 |12.09. |
|Ïðåîáðàçîâàíèÿ ãðàôèêîâ òðèãîíîìåòðè÷åñêèõ ôóíêöèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû? ¹508 [4) 0,5; 7) 1]; ¹517 [1) 2(; 5) 20(]; ãðàôèêè – íà äîñêå (ïî íåîáõîäèìîñòè); [à) 0,5; á) 5(; â) 4(; ã) [pic]; ä) 1,25(; å) 10]
2. Óñòíî: äàí ãðàôèê ôóíêöèè y = f(x) (íà äîñêå). Îáúÿñíèòå, êàê ïîëó÷èòü èç íåãî ãðàôèêè: à) y = f(x – a) + b; á) y = –f(–x); â) y = |f(x)|; ã) |y| = f(x); ä) |y| = |f(x)| (ïîêàçàòü). Êàêèå èç íèõ ÿâëÿþòñÿ ôóíêöèÿìè? [à) – â)].
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ïîñòðîéòå ãðàôèêè ôóíêöèé (âñïîìíèòü çàïèñè):
1) Â.: ñòð. 244, ¹539 (9); 2) y = sin(–x – [pic]); 3) Â.: ñòð. 254, ¹561 (11); 4) Â.: ñòð. 245, ¹540 [ã) – ïîðÿäîê!] 5) à) y = [tgx]; á) y = tg[x]; â) y = {tgx}; ã) y = tg{x}.
4. Ïîâòîðåíèå. Ïðèâåäèòå ê çíà÷åíèþ òðèãîíîìåòðè÷åñêîé ôóíêöèè íàèìåíüøåãî ïîëîæèòåëüíîãî àðãóìåíòà (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
à) sin20; á) cos30 [à) cos(6,5( – 20); á) sin(30 – 9,5()].
Äîìàøíåå çàäàíèå: ïîâòîðèòå ôîðìóëû ñëîæåíèÿ: f(( ( (), ãäå f – òðèãîíîìåòðè÷åñêèå ôóíêöèè); Â.: ¹568 (3, 7); ¹539 (6); ¹561 (6); ¹562; ¹569; 1) Äëÿ ôóíêöèè f(x) = cosx ïîñòðîéòå: à) [f(x)]; á) f([x]); â) {f(x)}; ã) f({x}); 2) Íàéäèòå îñíîâíîé ïåðèîä ôóíêöèè h(x) = 3tg1,75x + 5cos2x.
|Óðîê 12 |16.09. |
|Ãàðìîíè÷åñêèå êîëåáàíèÿ. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Ïåðèîä h(x)? [4(, òàê êàê ÍÎÊ ([pic]; 1) = 1]
Ãðàôèêè – çàãîòîâèòü íà äîñêå!
2. Íîâûé ìàòåðèàë. Òðèãîíîìåòðè÷åñêèå ôóíêöèè ÷àñòî èñïîëüçóþò äëÿ îïèñàíèÿ ðàçëè÷íûõ ôèçè÷åñêèõ ïðîöåññîâ, ñâÿçàííûõ ñ êîëåáàòåëüíûì äâèæåíèåì. Ðàññìîòðèì ïðîñòåéøèé èç ýòèõ ïðîöåññîâ, íàçûâàåìûé ãàðìîíè÷åñêèìè êîëåáàíèÿìè.
[pic]
Ïóñòü òî÷êà äâèæåòñÿ ïî îêðóæíîñòè ðàäèóñà R ñ ïîñòîÿííîé óãëîâîé ñêîðîñòüþ (. Âûáåðåì äåêàðòîâó ñèñòåìó êîîðäèíàò òàê, ÷òîáû åå íà÷àëî ñîâïàëî ñ öåíòðîì ýòîé îêðóæíîñòè. Çà ïðîìåæóòîê âðåìåíè t òî÷êà ïðîõîäèò ïóòü (t ïî îêðóæíîñòè. Åñëè Ð( – íà÷àëüíîå ïîëîæåíèå òî÷êè, òî ïî èñòå÷åíèè âðåìåíè t åå ïîëîæåíèå – Ð(t + ( (x; y) (ñì. ðèñ.). Òîãäà, x = R(cos((t + (); y = R(sin((t + (), ãäå x è y – êîîðäèíàòû òî÷êè â ìîìåíò âðåìåíè t.
Ïîëó÷åííûå óðàâíåíèÿ çàäàþò çàâèñèìîñòü êîîðäèíàò òî÷êè îò âðåìåíè. Çàâèñèìîñòè òàêîãî âèäà è íàçûâàþòñÿ ãàðìîíè÷åñêèìè êîëåáàíèÿìè, ïðè÷åì, òàê êàê cosx = sin([pic] + x), òî ïåðâîå óðàâíåíèå ìîæíî ïåðåïèñàòü òàê: x = R(sin((t + ( + [pic]), òî åñòü îáà óðàâíåíèÿ èìåþò îäèíàêîâûé âèä.  äàëüíåéøåì, óñëîâèìñÿ çàïèñûâàòü óðàâíåíèå ïðîèçâîëüíûõ ãàðìîíè÷åñêèå êîëåáàíèÿ â âèäå: f(t) = R(sin((t + (), ïðè÷åì A = |R| > 0 – àìïëèòóäà êîëåáàíèé (ïîêàçûâàåò íàèáîëüøåå çíà÷åíèå ôóíêöèè); ( – óãëîâàÿ ÷àñòîòà (ïîêàçûâàåò êîëè÷åñòâî ïîëíûõ êîëåáàíèé òî÷êè çà 2( åäèíèö âðåìåíè); ( – íà÷àëüíàÿ ôàçà êîëåáàíèé (ïîêàçûâàåò íà÷àëüíîå ïîëîæåíèå òî÷êè). Óñëîâèëèñü òàêæå, ÷òî ( > 0; (([0; 2(). Êðîìå òîãî, ðàññìàòðèâàåòñÿ âåëè÷èíà Ò = [pic]> 0 – ïåðèîä êîëåáàíèÿ (ïîêàçûâàåò âðåìÿ îäíîãî ïîëíîãî êîëåáàíèÿ).
Ïðèìåðû ôèçè÷åñêèõ âåëè÷èí, èçìåíÿþùèõñÿ ïî ýòîìó çàêîíó: îòêëîíåíèå îò ïîëîæåíèÿ ðàâíîâåñèÿ ãðóçà íà ïðóæèíå èëè íà íåâåñîìîé íèòè (ìàòåìàòè÷åñêèé ìàÿòíèê), íàïðÿæåíèå è ñèëà ïåðåìåííîãî òîêà è ïð.
Äëÿ ëþáûõ ãàðìîíè÷åñêèõ êîëåáàíèé ìîæíî îïðåäåëèòü èõ ïàðàìåòðû è ïîñòðîèòü ãðàôèê.
Ïðèìåð. f(t) = 3sin(–0,5t + [pic]). Ïðåîáðàçóåì: f(t) = –3sin(0,5t – [pic]) ( f(t) = –3sin(0,5t + [pic]).
Ïàðàìåòðû: À = 3; ( = 0,5; T = 4(; ( = [pic].
Ãðàôèê: f(t) = –3sin0,5(t – [pic]) (ïîêàçàòü íà äîñêå; ïàðàìåòðû – ñàìîêîíòðîëü!)
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Â.: ñòð. 247, ¹541 (3, 4).
Äîìàøíåå çàäàíèå: Â.: ñòð. 245 – 246, ï. 7; ¹541 (1, 2, 6); .ïîâòîðèòå ôîðìóëû äâîéíîãî, òðîéíîãî è ïîëîâèííîãî àðãóìåíòà; ¹568 (4; 7); ¹591 (4; 7); ¹596 (5); ñòð. 324, Ê_10, ¹7.
|Óðîê 13, 14 |17.09. |
|Ñëîæåíèå ãàðìîíè÷åñêèõ êîëåáàíèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
2. Óñòíî: 1) Íàéäèòå îñíîâíîé ïåðèîä ôóíêöèè f(x) = sin3x(cos[pic] – cos3x(sin[pic].
[f(x) = sin(3x – [pic]); T = [pic]]
2) Óïðîñòèòå: à) [pic]; á) [pic]; â) [pic].
[à) [pic]; á) sin(t – (), ãäå [pic]; â) [pic]]
3. Íîâûé ìàòåðèàë. 1) Ðàññìîòðèì óðàâíåíèå ïðîèçâîëüíîãî ãàðìîíè÷åñêîãî êîëåáàíèÿ
f(t) = R(sin((t + () è ïðåîáðàçóåì ïðàâóþ ÷àñòü: R(sin((t + () = R(sin(t(cos( + cos(t(sin() = Rcos((sin(t + Rsin((cos(t = R1sin(t + R2cos(t = R1sin(t + R2sin((t + [pic]), ãäå R1 = Rcos(; R2 = Rsin(.
Òàêèì îáðàçîì, ëþáîå ãàðìîíè÷åñêîå êîëåáàíèå ìîæíî ïðåäñòàâèòü â âèäå ñóììû äâóõ ãàðìîíè÷åñêèõ êîëåáàíèé ñ îäèíàêîâîé ÷àñòîòîé.
2) À) Ðàññìîòðèì ñóììó äâóõ ãàðìîíè÷åñêèõ êîëåáàíèé ñ îäèíàêîâîé ÷àñòîòîé è íóëåâîé íà÷àëüíîé ôàçîé: R1sin(t + R2cos(t = [pic] = Rsin((t + (), ãäå [pic]; [pic].
Á) Åñëè êîëåáàíèÿ èìåþò íåíóëåâûå íà÷àëüíûå ôàçû, òî R1sin((t + (1) + R2sin((t + (2) = (R1cos(1(+ R2cos(2)(sin(t + (R1sin(1(+ R2sin(2)cos(t = C1(sin(t + C2(cos(t, òî åñòü, ìû ïðèõîäèì ê ñëó÷àþ À). Ïðîäîëæèâ ïðåîáðàçîâàíèÿ ìîæíî â îáùåì âèäå ïîëó÷èòü ïàðàìåòðû ñóììû ãàðìîíè÷åñêèõ êîëåáàíèé. Æåëàþùèå – ïðî÷òóò â ó÷åáíèêå.
Òàêèì îáðàçîì, ñóììà ãàðìîíè÷åñêèõ êîëåáàíèé îäèíàêîâîé ÷àñòîòû åñòü ãàðìîíè÷åñêîå êîëåáàíèå òîé æå ÷àñòîòû.
Ïîëó÷åííûå ôàêòû ïîçâîëÿþò ïðè îïèñàíèè ôèçè÷åñêèõ ïðîöåññîâ ïðåäñòàâëÿòü ñëîæíûå ãàðìîíè÷åñêèå êîëåáàíèÿ â âèäå ñóììû äâóõ áîëåå ïðîñòûõ è íàîáîðîò, íàõîäèòü óðàâíåíèå ðåçóëüòèðóþùèõ êîëåáàíèé äëÿ âåëè÷èíû, ó÷àñòâóþùåé â äâóõ êîëåáàòåëüíûõ äâèæåíèÿõ ñ îäèíàêîâîé ÷àñòîòîé.
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
1) Â.: ñòð. 273, à) ¹607 (2; âûïèñàòü ïàðàìåòðû êîëåáàíèÿ)
[25sin(3t + ( – [pic]), ãäå [pic]; À = 25; ( = 3; T = [pic]; ( = ( – [pic]];
á) ¹608 (2; áåç òî÷åê ýêñòðåìóìà)
[13sin(2x – () = 13sin2(x – 0,5(), ãäå [pic]; (13; ãðàôèê!];
â) ¹609 (2; 4) [2) 2sin(3t – [pic]); 4) [pic]]
2) Íàéäèòå îñíîâíîé ïåðèîä è ýêñòðåìàëüíûå çíà÷åíèÿ ôóíêöèè:
à) [pic] [6sin[pic]; T = 2(2; (6]
á) [pic] [1 – 2cos4(, ( ( (n, ãäå n(Z; T = ( (îáëàñòü îïðåäåëåíèÿ!); max: 3; min: –1]
â) [pic], [pic] [[pic]; ôóíêöèÿ íå ÿâëÿåòñÿ ïåðèîäè÷åñêîé! max: íå ñóùåñòâóåò; min: 2 (îáîñíîâàíèÿ!)]
3) Ïîñòðîéòå ãðàôèê óðàâíåíèÿ: |y| = sin4|x| – cos4|x| [|y| = –cos2|x|; ãðàôèê!]
Ñëåäóþùèé óðîê – ê/ð (òåòðàäè)!
Äîìàøíåå çàäàíèå: Â.: ï. 7 (ñòð. 271, 272); ¹607 (3; âûïèñàòü ïàðàìåòðû êîëåáàíèÿ); ¹608 (4; áåç òî÷åê ýêñòðåìóìà); ¹609 (1; 3); ñòð. 324, Ê_11, ¹6. 1) Äàíà ôóíêöèÿ f(x) = tg2x – 3ctg0,8x; à) èññëåäóéòå åå íà ÷åòíîñòü; á) äîêàæèòå ïî îïðåäåëåíèþ, ÷òî 15( – åå ïåðèîä; â) íàéäèòå åå îñíîâíîé ïåðèîä. 2) Íàéäèòå îñíîâíîé ïåðèîä è ýêñòðåìàëüíûå çíà÷åíèÿ ôóíêöèè y = sinx(cosx(cos2x(cos4x.
|Óðîê 15, 16 |19.09. |
|Êîíòðîëüíàÿ ðàáîòà ¹1. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
2. Êîíòðîëüíàÿ ðàáîòà ¹1 (90 ìèíóò).
Îòâåòû è ðåøåíèÿ.
|I âàðèàíò. |II âàðèàíò. |
|¹1. ×åòíàÿ. |¹1. Íå÷åòíàÿ. |
|¹2. À) 5; [pic]; –5. Â) [pic]. |¹2. À) –1; 0; [pic]. Â) 12(. |
|¹3. y = 0,5sin4x, T = (, òàê êàê [pic], n(Z. |¹3. y = 0,5sin4x, T = (, òàê êàê [pic], n(Z. |
|max: 0,5 (ïðîâåðèòü, ÷òî äîñòèãàåòñÿ!). |min: –0,5 (ïðîâåðèòü, ÷òî äîñòèãàåòñÿ!). |
|¹4. [pic]; |¹4. [pic]; |
|A = 0,5; ( = 2; T = (; ( = [pic]. |A = 2; ( = 0,5; T = 4(; ( = [pic]. |
|¹6. Ïóñòü (T ( 0 | (x(D tg[x + T] = tg[x] ( [x + T] – [x] = (n, n(Z. Òàê êàê |¹6. Ïóñòü (T ( 0 | (x(D cos[x + T] = cos[x]. ( [x + T] = ( [x] + 2(n, n(Z. |
|÷èñëà ýòîãî âèäà ïðè n ( 0 íå ÿâëÿþòñÿ öåëûìè, òî (x(D [x + T] = [x], ÷òî |Òàê êàê ÷èñëà ýòîãî âèäà ïðè n ( 0 íå ÿâëÿþòñÿ öåëûìè, òî (x(D [x + T] = |
|íåâîçìîæíî. |([x], ÷òî íåâîçìîæíî. |
|¹7. Ïðèìåíèì òîæäåñòâî:[pic]. Äëÿ ýòîãî óìíîæèì è ðàçäåëèì ëåâóþ ÷àñòü íà sin2x (I âàðèàíò) èëè íà sinx (II âàðèàíò). |
|Óðîê 17 |23.09. |
|Ðåøåíèå óðàâíåíèé âèäà sint = m. |
1. Ðàçáîð ê/ð.
2. Íîâûé ìàòåðèàë. Ìû ïðèñòóïàåì ê ðåøåíèþ ïðîñòåéøèõ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. Îñîáåííîñòüþ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé ÿâëÿåòñÿ òî, ÷òî â áîëüøèíñòâå ñëó÷àåâ îíè èìåþò áåñêîíå÷íîå ìíîæåñòâî ðåøåíèé. Äëÿ çàïèñè ðåøåíèé íàì ïîòðåáóåòñÿ ââåñòè íîâûå ïîíÿòèÿ.
Îïðåäåëåíèå. (m([–1; 1] arcsinm = x | 1) x([pic]; 2) sinx = m.
Ïðèìåðû (îáîñíîâàíèÿ!). 1) arcsin0 = 0, òàê êàê 0([pic] è sin0 = 0; 2) arcsin1 = [pic];
3) arcsin[pic] = [pic]; 4) arcsin(–0,5) = [pic]; 5) arcsin(–1,5) íå ñóùåñòâóåò.
Âîïðîñû. 1) Ïî÷åìó m([–1; 1]? [Ìíîæåñòâî çíà÷åíèé ñèíóñà]
2) Ïî÷åìó x([pic]? Ìîæíî ëè áûëî âûáðàòü äðóãîé ïðîìåæóòîê? Êàêîé? [Ëþáîé ïðîìåæóòîê âîçðàñòàíèÿ ñèíóñà îò –1 äî 1 èëè óáûâàíèÿ ñèíóñà îò 1 äî –1]
3) Êàê ñâÿçàíû àðêñèíóñû ïðîòèâîïîëîæíûõ ÷èñåë? Ïî÷åìó?
[pic]
Èòàê, (m([–1; 1] arcsin(–m) = – arcsinm (ñì. ðèñ. 1).
Òåïåðü ðàññìîòðèì ðåøåíèå óðàâíåíèÿ sint = m (ñì. ðèñ. 1, äîïîëíèòü).
À) Åñëè |m| > 1, òî ðåøåíèé íåò.
Á) Åñëè m = 0, òî t = (k, k(Z.
Åñëè m = 1, òî t = [pic] + 2(k, k(Z.
Ðèñ. 1
Åñëè m = –1, òî t = –[pic] + 2(k, k(Z.
Â) Äëÿ îñòàëüíûõ m: t = arcsinm + 2(l, l(Z èëè t = ( – arcsinm + 2(n, n(Z.
Ïðèìåíèìà ëè ýòà ôîðìóëà äëÿ ÷àñòíûõ ñëó÷àåâ ïóíêòà Á)? [Äà, íî çàïèñü ïî íåé ìåíåå óäîáíà]
Ïðèìåðû (äâà âèäà çàïèñè: «â ñòðî÷êó» èëè «â ñòîëáèê»).
1) sint = [pic] ( t = arcsin[pic] + 2(l, l(Z èëè t = ( – arcsin[pic] + 2(n, n(Z (
t = [pic] + 2(l, l(Z èëè t = [pic] + 2(n, n(Z. Îòâåò: {[pic] + 2(l | l(Z} ( {[pic] + 2(n | n(Z}.
2) sinx = 0,3 ( x = arcsin0,3 + 2(l, l(Z èëè x = ( – arcsin0,3 + 2(n, n(Z.
Îòâåò: {arcsin0,3 + 2(l | l(Z} ( {( – arcsin0,3 + 2(n | n(Z}.
3) siny = [pic] ( y = arcsin[pic] + 2(l, l(Z èëè y = ( – arcsin[pic] + 2(n, n(Z (
y = – arcsin[pic] + 2(l, l(Z èëè y = ( + arcsin[pic] + 2(n, n(Z.
Îòâåò: {– arcsin[pic] + 2(l | l(Z} ( {( + arcsin[pic] + 2(n | n(Z}.
Ìîæíî ëè ïðåîäîëåòü íåóäîáñòâî çàïèñè ðåøåíèé óðàâíåíèÿ â âèäå îáúåäèíåíèÿ äâóõ ìíîæåñòâ? Îêàçûâàåòñÿ, ÷òî ìîæíî: t = (–1)karcsinm + (k, k(Z.
Ïðîâåðèì, ÷òî òàêàÿ ôîðìóëà çàäàåò òå æå ñàìûå ìíîæåñòâà ðåøåíèé [ïðè k = 2l – ïåðâîå ìíîæåñòâî, à ïðè k = 2n – 1 – âòîðîå]
Çàïèøåì, íàïðèìåð, ðåøåíèå óðàâíåíèÿ 3) äðóãèì ñïîñîáîì:
{(–1)k + 1(arcsin[pic] + (k | k(Z}
Äîìàøíåå çàäàíèå: Â.: ñòð. 281 – 285, ï. 1; ¹636 – ¹639; ¹7 èç ê/ð. Èíäèâèäóàëüíî (â çàâèñèìîñòè îò îøèáîê, äîïóùåííûõ â ê/ð): 1) (ïî âàðèàíòàì) íàéäèòå îñíîâíîé ïåðèîä è ýêñòðåìàëüíûå çíà÷åíèÿ ôóíêöèè [pic] | [pic]; 2) äëÿ ôóíêöèè f(x) = sinx ïîñòðîéòå: à) [f(x)]; á) f([x]); â) {f(x)}; ã) f({x}).
|Óðîê 18, 19 |24.09. |
|Ðåøåíèå óðàâíåíèé âèäà cost = m, tgt = m, ctgt = m. |
1. Ïðîâåðêà ä/ç: âîïðîñû? 1) [pic]. Ñëåäîâàòåëüíî, Ò = (; ýêñòðåìàëüíûå çíà÷åíèÿ: [pic] (äîñòèãàþòñÿ!); 2) èíäèâèäóàëüíî.
2. Íîâûé ìàòåðèàë. Äëÿ ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé, ñîäåðæàùèõ êîñèíóñ, íàì ïîòðåáóåòñÿ ââåñòè ïîíÿòèå àðêêîñèíóñà.
Îïðåäåëåíèå. (m([–1; 1] arccosm = x | 1) x([pic]; 2) cosx = m.
Ïðèìåðû (îáîñíîâàíèÿ!). 1) arccos0 = [pic], òàê êàê [pic]([pic] è cos[pic] = 0; 2) arccos(–1) = (;
3) arccos[pic] = [pic]; 4) arccos[pic] = [pic]; 5) arccos[pic] íå ñóùåñòâóåò.
Âîïðîñû. 1) Ïî÷åìó m([–1; 1]? [Ìíîæåñòâî çíà÷åíèé êîñèíóñà]
2) Ïî÷åìó x([pic]? Ìîæíî ëè áûëî âûáðàòü äðóãîé ïðîìåæóòîê? Êàêîé? [Ëþáîé ïðîìåæóòîê âîçðàñòàíèÿ êîñèíóñà îò –1 äî 1 èëè óáûâàíèÿ êîñèíóñà îò 1 äî –1]
[pic]
3) Êàê ñâÿçàíû àðêêîñèíóñû ïðîòèâîïîëîæíûõ ÷èñåë? Ïî÷åìó?
Èòàê, (m([–1; 1] arccos(–m) = ( – arccosm (ñì. ðèñ. 1).
Òåïåðü ðàññìîòðèì ðåøåíèå óðàâíåíèÿ cost = m (ñì. ðèñ. 1, äîïîëíèòü).
À) Åñëè |m| > 1, òî ðåøåíèé íåò.
Á) Åñëè m = 0, òî t = [pic] + (n, n(Z.
Åñëè m = 1, òî t = 2(n, n(Z.
Ðèñ. 1
Åñëè m = –1, òî t = ( + 2(n, n(Z.
Â) Äëÿ îñòàëüíûõ m: t = (arccosm + 2(n, n(Z.
Ïðèìåíèìà ëè ýòà ôîðìóëà äëÿ ÷àñòíûõ ñëó÷àåâ ïóíêòà Á)? [Äà, íî çàïèñü ïî íåé ìåíåå óäîáíà]
Îáðàòèòå âíèìàíèå, ÷òî óðàâíåíèå cost = m (òàêæå, êàê è óðàâíåíèå sint = m) ìîæíî áûëî èññëåäîâàòü, ïîëüçóÿñü ãðàôèêàìè òðèãîíîìåòðè÷åñêèõ ôóíêöèé, à íå åäèíè÷íîé îêðóæíîñòüþ!
Ïðèìåðû. 1) cost = = [pic] ( t = (arccos[pic] + 2(n, n(Z ( t = ((( – arccos[pic]) + 2(n, n(Z ( t = ([pic] + 2(n, n(Z. Îòâåò: {([pic] + 2(n | n(Z}
2) cosy = [pic] ( y = (arccos[pic] + 2(n, n(Z. ×òî íåîáõîäèìî áûëî ñäåëàòü, ïðåæäå ÷åì çàïèñûâàòü ðåøåíèÿ? [Ïðîâåðèòü, ÷òî [pic] < 1] Îòâåò: {(arccos[pic] + 2(n | n(Z}.
3) cosx = –0,2 ( x = (arccos(–0,2) + 2(n, n(Z ( x = ((( – arccos0,2) + 2(n, n(Z.
Îòâåò: {((( – arccos0,2) + 2(n, n(Z | n(Z}.
Ïî àíàëîãèè ñ àðêñèíóñîì è àðêêîñèíóñîì ïîïðîáóéòå ñôîðìóëèðîâàòü îïðåäåëåíèÿ àðêòàíãåíñà è àðêêîòàíãåíñà.  ÷åì ðàçíèöà? [Å(tgx) = E(ctgx) = R]
Îïðåäåëåíèå. 1) arctgm = x | 1) x([pic]; 2) tgx = m.
2) arcctgm = x | 1) x([pic]; 2) ctgx = m.
Ïðèìåðû (îáîñíîâàíèÿ!). 1) arctg0 = 0, òàê êàê 0([pic] è tg0 = 0; 2) arcctg0 = [pic]; 3) arctg(–1) = [pic]; 4) arcctg(–1) = [pic]; 5) arctg[pic] = [pic]; 6) arcctg[pic] = [pic].
Êàê ñâÿçàíû à) àðêòàíãåíñû; á) àðêêîòàíãåíñû ïðîòèâîïîëîæíûõ ÷èñåë? Ïî÷åìó?
[pic]
(m(R 1) arctg(–m) = – arctgm; 2) arcctg(–m) = ( – arcctgm (ïîêàçàòü íà ãðàôèêàõ ôóíêöèé tgx è ctgx).
Ðàññìîòðèì ðåøåíèÿ óðàâíåíèé tgt = m è ctgt = m. (m(R òàêèå óðàâíåíèÿ èìåþò ðåøåíèÿ, ïðè÷åì íà êàæäîì ïðîìåæóòêå äëèíû ( – åäèíñòâåííîå (ïîêàçàòü íà ãðàôèêàõ). Èç êàêèõ ñâîéñòâ ôóíêöèé ýòî ñëåäóåò? [Ìíîæåñòâî çíà÷åíèé; ìîíîòîííîñòü]
Ðèñ. 2
Ïîýòîìó, íåò ñìûñëà ðàññìàòðèâàòü ÷àñòíûå ñëó÷àè, à ìîæíî ñðàçó, ó÷èòûâàÿ ïåðèîäè÷íîñòü ýòèõ ôóíêöèé, çàïèñàòü ìíîæåñòâà ðåøåíèé: {arctgm + (n | n(Z} è {arcñtgm + (n | n(Z} ñîîòâåòñòâåííî.
Òå æå ðåçóëüòàòû ìîæíî áûëî ïîëó÷èòü, ðàññìàòðèâàÿ íå ãðàôèêè, à îñè òàíãåíñîâ è êîòàíãåíñîâ íà åäèíè÷íîé îêðóæíîñòè (ñì. ðèñ. 2).
Ïðèìåðû. 1) tgt = 1 ( t = arctg1 + (n, n(Z ( t = [pic]. Îòâåò: {[pic] + (n | n(Z}
2) tgx = –3,6 ( t = arctg(–3,6) + (n, n(Z ( t = –arctg3,6 + (n, n(Z.
Îòâåò: {–arctg3,6 + (n | n(Z}
3) ctgy = –[pic] ( y = arcctg(–[pic]) + (n, n(Z ( ( –arcctg[pic] + (n, n(Z ( y = [pic] + (n, n(Z. Îòâåò: {[pic] + (n | n(Z}
Íàéäèòå äðóãîé ñïîñîá ðåøåíèÿ ýòîãî óðàâíåíèÿ [ctgy = –[pic] ( tgy = –[pic] ( y = –arctg[pic] + (n, n(Z ( y = –[pic] + (n, n(Z] Ïî÷åìó ïîëó÷èëèñü ðàçíûå îòâåòû? [Îòâåò – îäèí è òîò æå, íî çàïèñàí ðàçíûìè ñïîñîáàìè!]
Ïðè ðàçëè÷íûõ ñïîñîáàõ ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé ÷àñòî ïîëó÷àþòñÿ îòâåòû, çàïèñàííûå ïî ðàçíîìó, ïðè÷åì íå âñåãäà ýòî âèäíî ñðàçó!
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé è ïðîâåðêîé íà äîñêå): Ðåøèòå óðàâíåíèÿ:
1) à) cost = [pic]; á) tgt = –[pic]; â) ñtgt = –[pic], ãäå m > 0.
[à) ïðè m ( 1 ðåøåíèé íåò; ïðè m = 1 {2(n | n(Z}; á) –arctg[pic] + (n, n(Z; â) ( – arcñtg[pic] + (n, n(Z]
2) Â.: ñòð. 294, ¹655 (2; 4; 14) [2) [pic], n(Z; 4) [pic], n(Z ëèáî [pic], k(Z (â çàâèñèìîñòè îò ñïîñîáà ðåøåíèÿ); 14) (–1)n[pic]arcsin0,4 + [pic], n(Z]
Äîìàøíåå çàäàíèå: Â.: ñòð. 286 – 290, ïï. 2 – 3; ¹641; ¹643; ¹645; ¹646; ¹648; ¹650 (1 – 3); ¹652. 1) Ðåøèòå óðàâíåíèÿ: à) cost = [pic]; á) cosx = [pic], ãäå à > 0; b > 0. 2) Âû÷èñëèòå: [pic].
|Óðîê 20, 21 |26.09. |
|Îñíîâíûå ìåòîäû ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
2. Óñòíî: Ñóùåñòâóþò ëè ÷èñëà (îáîñíîâàíèÿ!): à) arcsin([pic]); á) p | arcsinp = [pic]; â) arccos(; ã) p | arccosp = (; ä) arctg100; å) p | arctgp = 100; æ) arcctg[pic]; ç) p | arcctgp = [pic]? [à) íåò; á) äà; â) íåò; ã) äà; ä) äà; å) íåò; æ) äà; ç) äà]
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì îñíîâíîé ìåòîä ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. Ýòî – ñïîñîá çàìåíû ïåðåìåííûõ (ïîäñòàíîâêè). Îí ïðèìåíÿåòñÿ äëÿ óðàâíåíèé âèäà f(g(t) = m, ãäå õîòÿ áû îäíà èç ôóíêöèé – òðèãîíîìåòðè÷åñêàÿ.
Ïðèìåðû. 1) 2cos(3t + [pic]) = [pic]; x = 3t + [pic]; cosx = [pic] ( x = ([pic] + 2(n, n(Z; t = ([pic] – [pic] + [pic], n(Z. Îòâåò ìîæíî çàïèñàòü è ïî äðóãîìó. Îòâåò: {– [pic] + [pic] | n(Z} ( {[pic] k(Z}.
 êîíöå ïðîøëîãî óðîêà ìû ðåøàëè áîëåå ïðîñòûå óðàâíåíèÿ òàêîãî æå âèäà, ãäå çàìåíà ïåðåìåííûõ áûëà «íåÿâíîé»!
2) 2sin3x + 3sin2x = 2sinx; sinx = y; 2y3 + 3y2 = 2y ( y(2y2 + 3y – 2) = 0 ( y = 0 èëè y = 0,5 èëè y = –2; x = (k, k(Z èëè x = (–1)n[pic] + (n, n(Z.
3) |sinx(cos3x + cosx(sin3x| = 1 ( |sin4x| = 1; t = 4x; |sint| = 1 ( t = [pic] + (n, n(Z (åäèíè÷íàÿ îêðóæíîñòü!); x = [pic], n(Z.
Îáðàòèòå âíèìàíèå, ÷òî ðàçáèâàòü óðàâíåíèå ñ ìîäóëåì íà ñîâîêóïíîñòü äâóõ óðàâíåíèé íåðàöèîíàëüíî!
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé è ïðîâåðêîé íà äîñêå): 1) Â.: ñòð. 294, ¹656 (17) [[pic], n(Z èëè [pic], k(Z]
sin0,3x(cos0,3x = –0,5 [[pic], n(Z]
2) Â.: ñòð. 294, ¹655 (8; 15) [8) [pic], n(Z;15) (3arccos[pic] – [pic] + 6(n, n(Z]
[pic] [(–1)k[pic] + [pic]+ [pic], k(Z]
3) Â.: ñòð. 294, ¹655 (9; 10; 13; 19) [9) [pic], n(Z; ìîæíî ðåøàòü òàêæå ñ ïîìîùüþ ôîðìóëû ïîíèæåíèÿ ñòåïåíè! 10) (10( + 20( + 60(n, n(Z; 13) (; [pic], n(N]
cos4([pic]) + cos2([pic]) = 2 [n2, n(Z+]
Äîìàøíåå çàäàíèå: Â.: ñòð. 291 –294, ï. 4 (ïðèìåðû 1, 2, 6 – 8); ¹655 (5; 6; 7; 11; 16; 18); ¹656 (16); ðåøèòå óðàâíåíèå x2 – 2xcos2(x + 1 = 0.
|Óðîê 22 |30.09. |
|Îñíîâíûå ìåòîäû ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
|Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹2. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
Ðàññìîòðèì ðåøåíèå íàèáîëåå «íåïðèÿòíîãî» êëàññà óðàâíåíèé.
2. Ïèñüìåííî (íà äîñêå è â òåòðàäÿõ):
à) sin(cos0,1x) = 0,5; á) tg(sin(2x)) = –2; â) sin((cos2x) = 1
[à) cos0,1x = [pic], n(Z èëè cos0,1x = [pic], m(Z; îòáîð! x = [pic], k(Z; á) sin(2x) = –arctg2 + (n, n(Z; [pic] (ãðàôèê èëè åäèíè÷íàÿ îêðóæíîñòü); (; â) cos2x = [pic], n(Z; îòáîð! x = [pic], k(Z]
Äîìàøíåå çàäàíèå: Â.: ¹655 (12; 17); ¹656 (18); ðåøèòå óðàâíåíèÿ: 1) |cos10[pic]| = [pic]; 2) tg(sin5t) = –1; 3) cos([pic]sin(y) = [pic]; äîï. ê ñ/ð.
3. Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹2 (íà ëèñòî÷êàõ; 20 ìèíóò).
Îòâåòû.
|I âàðèàíò. |II âàðèàíò. |
|¹1. [pic]. |¹1. [pic]. |
|¹2. à) 2(n, n(Z; á) [pic], n(Z; |¹2. à) [pic], n(Z; á) [pic], n(Z; â) [pic], n(Z+. |
|â) [pic], n(Z+. | |
|¹3. Ïðè à([cos1; 1] x = (–1)karcsin((arccosa) + (n, n(Z; ïðè îñòàëüíûõ à – |¹3. Ïðè à([–sin1; sin1] x =(arcños(arcsina) + 2(n, n(Z; ïðè îñòàëüíûõ à – |
|ðåøåíèé íåò. |ðåøåíèé íåò. |
|Óðîê 23, 24 |1.10. |
|×àñòíûå ìåòîäû ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
1. Ðàçáîð ñ/ð.
2. Ïðîâåðêà ä/ç: âîïðîñû?
3. Óñòíî: 1) Ðåøèòå â öåëûõ ÷èñëàõ óðàâíåíèå: 4n = 2k + 1 [(];
2) Ïðè êàêèõ öåëûõ m óðàâíåíèå (x2 – x + m = 0 èìååò êîðíè? [k(Z–];
3) Ðåøèòå óðàâíåíèÿ: à) [pic]; á) [pic] [à) 6; á) –1; 2].
4. Íîâûé ìàòåðèàë. Ñèòóàöèè, àíàëîãè÷íûå çàäàíèþ 3), ÷àñòî ñêëàäûâàþòñÿ è ïðè ðåøåíèè òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. Ðàññìîòðèì ïðèìåðû.
1) (1 – sinx)(tg2x – 3) = 0 ( [pic] ( [pic] (åäèíè÷íàÿ îêðóæíîñòü!).
2) [pic] ( [pic] ( [pic] ( [pic] èëè [pic] (åäèíè÷íàÿ îêðóæíîñòü!).
3) tg0,1x = tg0,9x ( [pic] ( [pic] ( [pic]; îòáîð êîðíåé:
[pic] ( [pic]; [pic] ( [pic]( [pic].
Îòâåò: [pic].
5. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå): Ðåøèòå óðàâíåíèÿ:
1) cos6x(ctgx = ctgx [[pic](åäèíè÷íàÿ îêðóæíîñòü)];
2) [pic] [[pic](åäèíè÷íàÿ îêðóæíîñòü)];
3) cosx(cos2x(cos4x = [pic] [Ïðîâåðèòü, ÷òî ÷èñëà âèäà (n | n(Z íå ÿâëÿþòñÿ êîðíÿìè äàííîãî óðàâíåíèÿ, çàòåì, óìíîæèòü è ðàçäåëèòü ëåâóþ ÷àñòü íà sinx! [pic] [pic]]
4) Â.: ñòð. 294, ¹656 (12; 7; 6)
[12) [pic]; 7) [pic]; [pic]; 6) [pic]; [pic]]
Äîìàøíåå çàäàíèå: Â.: ¹656 (1 – 5; 8; 11; 14); ðåøèòå óðàâíåíèå: [pic]. Ïîâòîðèòå ôîðìóëû ïîíèæåíèÿ ñòåïåíè.
|Óðîê 25, 26 |3.10. |
|×àñòíûå ìåòîäû ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Îòâåò â ïðîäèêòîâàííîì óðàâíåíèè? [[pic]]
2. Óñòíî: 1) Íàéäèòå [pic], åñëè a2 – 4ab + 3b2 = 0; êàê íàçûâàþòñÿ òàêèå óðàâíåíèÿ?
[Åñëè b ( 0, òî 1 èëè 3; îäíîðîäíûå];
2) Ïîíèçüòå ñòåïåíü âûðàæåíèÿ: 10sin25x [5(1 – cos10x)];
3) Âûðàçèòå cos6t – sin6t ÷åðåç cos2t è sin2t [cos2t(1 – 0,25sin2t)].
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì ïðèìåðû òðèãîíîìåòðè÷åñêèõ óðàâíåíèé, êîòîðûå ñâîäÿòñÿ ê êâàäðàòíûì óðàâíåíèÿì ñ ïîìîùüþ òðèãîíîìåòðè÷åñêèõ ïðåîáðàçîâàíèé.
1) 4sinx = 4sin2x + 3cos2x ( 4sinx = 4sin2x + 3(1 – sin2x); sinx = y; y2 – 4y + 3 = 0 ( y = 1 èëè y = 3; x = [pic]. Ïî÷åìó âûðàæàëè cos2x, à íå sin2x?
2) cos2x – 3sinx(cosx = – 1 ( sin2x – 3sinx(cosx + 2cos2x = 0; ýòî îäíîðîäíîå òðèãîíîìåòðè÷åñêîå óðàâíåíèå! cosx ( 0, òàê êàê åñëè cosx = 0, òî è sinx = 0, ÷òî îäíîâðåìåííî íåâîçìîæíî; tg2x – 3tgx + 2 = 0 ( tgx = 1 èëè tgx = 2 ( x = [pic] èëè x = [pic]. Ìîæíî ëè áûëî äåëèòü âñå ÷ëåíû óðàâíåíèÿ íà sin2x? [Äà, è ïîëó÷èòü êâàäðàòíîå óðàâíåíèå îòíîñèòåëüíî ctgx]
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðåøèòå óðàâíåíèÿ:
1) 3 – 3sin2x – 3cosx = 0 [cosx = 0 èëè cosx = 1; [pic]];
2) Â.: ñòð. 297, ¹658 (3) è íàéäèòå êîðíè ýòîãî óðàâíåíèÿ, ëåæàùèå íà [pic]
[y = sinx; 4y3 – 4y2 – y + 1 = 0; y = 1 èëè y = (0,5; [pic]; ïîêàçàòü äâà ñïîñîáà: åäèíè÷íàÿ îêðóæíîñòü èëè äâîéíûå íåðàâåíñòâà; [pic] ];
3) tgx – tg[pic] [tg2x – tgx – 1 = 0 è [pic]; [pic]];
4) Â.: ñòð. 297, ¹657 (2; 7; 9; 10) [2) äâà ñïîñîáà; [pic]]; 7) [pic]; 9) Ìîæíî ëè ðàçäåëèòü äàííîå óðàâíåíèå ïî÷ëåííî íà cos4x? [pic]; 10) Ïîíèæåíèå ñòåïåíè! [pic]; ïðè ñâåäåíèè ê îäíîðîäíîìó óðàâíåíèþ âòîðîé ñòåïåíè: [pic]];
5) [pic] [Ñóììà êóáîâ! |sin2x| = [pic]; [pic]].
Ñëåäóþùèé óðîê – ñ/ð!
Äîìàøíåå çàäàíèå: Â.: ï. 5 (ñòð. 295 – 297); ¹656 (19; 13; 15); ¹657 (1; 5; 6; 11); ¹658 (2; 4; 6). Ïîâòîðèòå ôîðìóëû ñóììû, ðàçíîñòè è ïðîèçâåäåíèÿ òðèãîíîìåòðè÷åñêèõ ôóíêöèé.
|Óðîê 27, 28 |7.10. |
|×àñòíûå ìåòîäû ðåøåíèÿ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
|Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹3. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
Ðàññìîòðèì óðàâíåíèÿ, â êîòîðûõ ôîðìóëû ñóììû è ðàçíîñòè òðèãîíîìåòðè÷åñêèõ ôóíêöèé ïîçâîëÿþò ðàçëîæèòü ëåâóþ ÷àñòü íà ìíîæèòåëè.
2. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Â.: ñòð. 298, ¹659 (5; 11; 8; 7) [5) sin5x = 0 èëè cos4x = –0,5; [pic]; 11) 2sin1,5x(sin0,5x = 2sin1,5x(cos1,5x ( sin1,5x = 0 èëè sin[pic] = 0 èëè cos[pic] = 0; [pic]; 8) [pic] ( [pic]; [pic]; 7) [pic]; ïðè a = b = 0 x(R; ïðè a = 0, b ( 0 [pic]; ïðè a ( 0, b ( 0 [pic]]
3. Íîâûé ìàòåðèàë. Åùå îäèí âèä óðàâíåíèé, êîòîðûå âñòðå÷àþòñÿ äîñòàòî÷íî ÷àñòî: asinx + bcosx = c, ãäå {a; b; c}(R, ïðè÷åì a2 + b2 ( 0. Ïðè ñ = 0 òàêîå óðàâíåíèå ÿâëÿåòñÿ îäíîðîäíûì (I ñòåïåíè). À êàê åãî ðåøàòü, åñëè ñ ( 0? Íàèáîëåå åñòåñòâåííûé ñïîñîá – èñïîëüçîâàòü ôîðìóëó ñëîæåíèÿ ãàðìîíè÷åñêèõ êîëåáàíèé (ôîðìóëó äîïîëíèòåëüíîãî àðãóìåíòà), òî åñòü, ïðèâåñòè óðàâíåíèå ê âèäó: [pic], ãäå [pic]. Ïîëó÷åííîå óðàâíåíèå ðàâíîñèëüíî óðàâíåíèþ [pic]. Îíî èìååò ðåøåíèÿ ò. è ò. ò., êîãäà [pic]. Ñóùåñòâóåò è äðóãèå ñïîñîáû ðåøåíèÿ òàêèõ óðàâíåíèé, îäèí èç êîòîðûõ âû ïðî÷èòàåòå äîìà â ó÷åáíèêå, à äðóãîé ïîïðîáóåòå ïðèäóìàòü ñàìè.
4. Ïèñüìåííî (íà äîñêå è â òåòðàäÿõ):
Â.: ñòð. 299, ¹661 (2; 5)
[2) [pic]; 5) [pic]]
Äîìàøíåå çàäàíèå: Â.: ï. 6 (ñòð. 298 – 299) è ïðèäóìàòü åùå îäèí ñïîñîá ðåøåíèÿ; ¹656 (20); ¹659 (3; 6; 10; 4); ¹661 (3; 4); ðåøèòå óðàâíåíèÿ: à) 1 + sin2x = 2cosx + sinx; á) 2cos[pic] = 1 – [pic]sinx. Äîï. ê ñ/ð.
5. Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹3 (íà ëèñòî÷êàõ; 25 ìèíóò).
Îòâåòû.
|I âàðèàíò. |II âàðèàíò. |
|¹1. [pic] = [pic]; 0; [pic]. |¹1. [pic] = [pic]; [pic]; [pic]; [pic]. |
|¹2. à) [pic]; |¹2. à) [pic]; |
|á) [pic]; â) (. |á) [pic]; â) (. |
|Óðîê 29, 30 |8.10. |
|Ðåøåíèå òðèãîíîìåòðè÷åñêèõ óðàâíåíèé, ñîäåðæàùèõ ìîäóëè. |
1. Ðàçáîð ñ/ð.
2. Ïðîâåðêà ä/ç: âîïðîñû? Êàêîé åùå ñïîñîá âîçìîæåí ïðè ðåøåíèè óðàâíåíèé âèäà asinx + bcosx = c? [Âûðàçèòü ÷åðåç ñèíóñ è êîñèíóñ ïîëîâèííîãî àðãóìåíòà: 2àsin0,5x(cos0,5x + bcos20,5x – bsin20,5x = ccos20,5x + csin20,5x; óðàâíåíèå ñâåäåòñÿ ê îäíîðîäíîìó è, â îòëè÷èå îò óíèâåðñàëüíîé ïîäñòàíîâêè, íå ïîòðåáóåòñÿ äåëàòü ïðîâåðêè!]
Ðåøèì íåñêîëüêî òðèãîíîìåòðè÷åñêèõ óðàâíåíèé, ïðèìåíÿÿ ñòàíäàðòíûå ïðèåìû äëÿ èõ ïðåîáðàçîâàíèé.
3 Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðåøèòå óðàâíåíèÿ:
1) cos2x + cos22x + cos23x = 1 è íàéäèòå âñå ðåøåíèÿ, ïðèíàäëåæàùèå [3; 5] [Ïîíèæåíèå ñòåïåíè; cosx(cos2x(cos3x = 0; [pic]; [pic] (åäèíè÷íàÿ îêðóæíîñòü)];
2) cos5x + cos3x = sin8x [Ôîðìóëû ñëîæåíèÿ è äâîéíîãî àðãóìåíòà; cos4x(sin(2,5x – [pic])(cos(1,5x + [pic]) = 0; [pic]];
3) sin6x + cos4x = 1 – 6sinx(cosx [Ôîðìóëû ñèíóñà òðîéíîãî è äâîéíîãî àðãóìåíòîâ; sin2x = 0 èëè sin2x = 1 èëè sin2x = –1,5; [pic]].
4 Óñòíî: èñïîëüçóÿ åäèíè÷íóþ îêðóæíîñòü (íà äîñêå), ðåøèòå óðàâíåíèÿ:
à) [pic]; á) cos|x| = 1; â) sin2x = 0,5; ã) sin|x| = –1.
[à) [pic]; á) [pic]; â) [pic]; ã) [pic]].
5. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
1) Íàéäèòå ðåøåíèÿ óðàâíåíèÿ 3sin2x + 3cosx + cos2x = 1, óäîâëåòâîðÿþùèå íåðàâåíñòâó sinx > 0 [cosx = 2 èëè [pic]; [pic] (åäèíè÷íàÿ îêðóæíîñòü)];
Ðåøèòå óðàâíåíèÿ:
2) [pic] [[pic](åäèíè÷íàÿ îêðóæíîñòü)];
3) |cosx|(cosx + sin2x = 0,5 [[pic];[pic] (åäèíè÷íàÿ îêðóæíîñòü)];
4) 6sin2x = 8|sinx| – 2 [|sinx| = 1 èëè |sinx| = [pic]; [pic] (åäèíè÷íàÿ îêðóæíîñòü)];
5) |sinx| = |cosx| [|tgx| = 1; [pic] (ãðàôèê)];
6) |sint| + |cost| = 1,4 [Âîçâåäåíèå â êâàäðàò! [pic]];
7) |sint – cost| = 1 – sin2t [|sint – cost| = |sint – cost|2; |sint – cost| = y; y = 0 èëè y = 1; [pic]].
Äîìàøíåå çàäàíèå: Â.: ï. 7 (ñòð. 299 – 300); ¹659 (12); ¹662 (2); ñòð. 324, K_12, ¹3. Ðåøèòå óðàâíåíèÿ: 1) 3sinx = 2cos2x, åñëè tgx < 0; 2) |sinx|(cosx = 0,5; 3) 4cos2x = 1 – 3|cosx|; 4) |tgx| = |sinx|; 5) [pic]; 6) sin2x – cos23x = 2|sin3x| + |sinx| – 2,25.
|Óðîê 31, 32 |10.10. |
|Ïðèìåíåíèå ñâîéñòâ òðèãîíîìåòðè÷åñêèõ ôóíêöèé ïðè ðåøåíèè óðàâíåíèé. Îáîáùàþùèé óðîê ïî ðåøåíèþ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
Ðàññìîòðèì òðèãîíîìåòðè÷åñêèå óðàâíåíèÿ, äëÿ ðåøåíèÿ êîòîðûõ, ïîìèìî ôîðìóë è ñòàíäàðòíûõ ïðèåìîâ, ïðèìåíÿþòñÿ ñâîéñòâà òðèãîíîìåòðè÷åñêèõ ôóíêöèé.
2. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðåøèòå óðàâíåíèÿ:
1) tgx + ctgx = [pic] [(x ( [pic], n(Z |tgx + ctgx| ( 2; (];
2) a) cosx + cos3x = 2 [[pic]; äâà ñòàíäàðòíûõ ñïîñîáà íàõîäèòü ïåðåñå÷åíèå ìíîæåñòâ; {2(n | n(Z}];
á) Íàéäèòå à(R | óðàâíåíèå cosx + cosax = 2 èìååò îäèí êîðåíü.
[(a(R x = 0 – êîðåíü óðàâíåíèÿ; åñëè à ( 0, òî n = ka, ãäå {n, k}(Z. a(Q];
3) sin3x + cos2,4x = 2 [[pic] ( [pic]; ïåðåñå÷åíèå ìíîæåñòâ ìîæíî íàõîäèòü ëèáî ðåøàÿ óðàâíåíèå, ëèáî èñïîëüçóÿ îñíîâíîé ïåðèîä ôóíêöèè f(x) = sin3x + cos2,4x: ÍÎÊ ([pic]; [pic]) = ÍÎÊ ([pic]; [pic]) = [pic] = [pic]; T = [pic]; [pic]];
4) sin2x(3sin2x – cos0,5x) = cos2x(2 + sin0,5x – 3cos2x) [2cos2x + sin2,5x = 3; [pic]; {( + 4(k | k(Z}];
5) cos((x) + x–1 = 0, x([0; 1] [Ôóíêöèÿ f(x) = cos((x) óáûâàåò íà äàííîì ïðîìåæóòêå, à ôóíêöèÿ g(x) = [pic] – âîçðàñòàåò; {1}];
6) 12sinx + 5cosx – 15 = 2x2 – 4x [13sin(x + t) = 2(x – 1)2 + 13, ãäå t = arccos[pic]; [pic]; (];
7) x2 – 2xcos(2(x) + 1 = 0 [D’ = cos2(2(x) – 1 ( 0 ( |cos(2(x)| = 1 ( x = 0,5n, n(Z; {1}];
8) Íàéäèòå à(R | óðàâíåíèå 2x2 – b(tg(cosx) + b2 = 0 èìååò îäèí êîðåíü.
[f(x) = 2x2 – b(tg(cosx) + b2 – ÷åòíàÿ, ïîýòîìó, åñëè êîðåíü åäèíñòâåííûé, òî x = 0; 0 ÿâëÿåòñÿ êîðíåì, åñëè b = 0 èëè b = tg1; ïðè ýòèõ b 0 – åäèíñòâåííûé êîðåíü (îöåíêà, èñïîëüçóþùàÿ âîçðàñòàíèå òàíãåíñà!)].
 çàêëþ÷åíèå, ðàññìîòðèì óðàâíåíèÿ, äëÿ ðåøåíèÿ êîòîðûõ ïðèìåíÿþòñÿ èñêóññòâåííûå ïðèåìû, õàðàêòåðíûå äëÿ òðèãîíîìåòðèè.
9) Â.: ñòð. 300, ¹662 (3). Ïî÷åìó íå õî÷åòñÿ èñïîëüçîâàòü ôîðìóëû òðîéíîãî àðãóìåíòà?
[(1 – sin2x)(cosx(sin3x + (1 – cos2x)(sinx(cos3x = [pic]; sin4x – sinx(cosx(cos2x = [pic]; sin4x = 1;
[pic]];
10) cos2x + cos4x + cos6x + cos8x = –0,5. Ïî÷åìó áåññìûñëåííî ãðóïïèðîâàòü è ñêëàäûâàòü êîñèíóñû? [à) sinx = 0 ( x = (n, n(Z – íå ÿâëÿåòñÿ ðåøåíèåì äàííîãî óðàâíåíèÿ; á) sinx ( 0, òîãäà, óìíîæèâ è ðàçäåëèâ íà sinx, ïîëó÷èì: [pic]; [pic]].
Ñëåäóþùèé óðîê – ê/ð!
Äîìàøíåå çàäàíèå: Â.: ñòð. 324, K_12 ¹4, ¹6. Ðåøèòå óðàâíåíèÿ: 1) cos5x + sin7,5x = –2; 2) |tg2x + ctg2x| = [pic]; 3) 2cos8x + cos2x + [pic]sin2x = 0; 4) [pic]; 5) 5sin2x – 3cos2x = |sin2x|; 6) sinx + cos[pic] + tg3x(sin[pic] = 0; 7) [pic], ãäå y([–3(; 2(].
|Óðîê 33, 34 |14.10. |
|Êîíòðîëüíàÿ ðàáîòà ¹2. |
Ðàçäàòü âîïðîñû ê çà÷åòó ïî òðèãîíîìåòðèè.
1. Êîíòðîëüíàÿ ðàáîòà ¹2 (90 ìèíóò).
Îòâåòû è óêàçàíèÿ.
|I âàðèàíò. |II âàðèàíò. |
|¹1. à) [pic]; |¹1. à) [pic]; |
|á) [pic]. |á) [pic]. |
|¹2. [pic] |¹2. [pic] |
|¹3. [pic]. |¹3. [pic]. |
|¹4. Ðåøåíèé íåò. ¹5. [pic]. |¹4. Ðåøåíèé íåò. ¹5. [pic]. |
|¹6. [pic]. ¹7. [pic]. |¹6. [pic]. ¹7. [pic] |
|¹8. [pic]. |¹8. [pic]. |
|¹9. [pic]. x = ( ÿâëÿåòñÿ êîðíåì óðàâíåíèÿ (ïðîâåðêà); íà[pic]íåò äðóãèõ |¹9. [pic]. x = [pic] ÿâëÿåòñÿ êîðíåì óðàâíåíèÿ (ïðîâåðêà); íà [0; (] íåò |
|ðåøåíèé, òàê êàê ôóíêöèÿ â ëåâîé ÷àñòè óðàâíåíèÿ óáûâàåò, à ôóíêöèÿ â ïðàâîé |äðóãèõ ðåøåíèé, òàê êàê ôóíêöèÿ â ëåâîé ÷àñòè óðàâíåíèÿ óáûâàåò, à ôóíêöèÿ â |
|÷àñòè óðàâíåíèÿ – âîçðàñòàåò; x < [pic] – íå ðåøåíèÿ, òàê êàê x – ( < [pic]< |ïðàâîé ÷àñòè óðàâíåíèÿ – âîçðàñòàåò; x < 0 – íå ðåøåíèÿ, òàê êàê x – [pic] < |
|–1; x > [pic] – íå ðåøåíèÿ, òàê êàê x – ( > [pic] > 1. |[pic]( < –1; x > ( – íå ðåøåíèÿ, òàê êàê x – [pic] > [pic] > 1. |
|¹10. a = 0 èëè a = 2sin1. ×åòíîñòü ôóíêöèè â ëåâîé ÷àñòè óðàâíåíèÿ è ïðîâåðêà|¹10. b = ctg1. ×åòíîñòü ôóíêöèè â ëåâîé ÷àñòè óðàâíåíèÿ è ïðîâåðêà |
|äîñòàòî÷íîñòè óñëîâèÿ. |äîñòàòî÷íîñòè óñëîâèÿ. |
|Óðîê 35, 36 |15.10. |
|Äîêàçàòåëüñòâî òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. |
1. Ðàçáîð ê/ð.
 9 êëàññå ìû óæå ðàññìàòðèâàëè íåêîòîðûå òðèãîíîìåòðè÷åñêèå íåðàâåíñòâà. Îñíîâíûå ìåòîäû èõ äîêàçàòåëüñòâà: èñïîëüçîâàíèå åäèíè÷íîé îêðóæíîñòè èëè ìîíîòîííîñòè òðèãîíîìåòðè÷åñêèõ ôóíêöèé íà îòäåëüíûõ ïðîìåæóòêàõ; ïðèìåíåíèå òîæäåñòâåííûõ òðèãîíîìåòðè÷åñêèõ ïðåîáðàçîâàíèé è ñòàíäàðòíûõ àëãåáðàè÷åñêèõ íåðàâåíñòâ.
2. Óñòíî: Äîêàæèòå íåðàâåíñòâà: 1) (x(R sinx(cosx ( 0,5 [ôîðìóëà äâîéíîãî àðãóìåíòà];
2) (x ( [pic], n(Z |tgx + ctgx| ( 2 [ñóììà âçàèìíî îáðàòíûõ ÷èñåë];
3) (t(R |sint| + |cost| ( 1 [åäèíè÷íàÿ îêðóæíîñòü];
4) (x(R |sinx + cosx| ( [pic] [ôîðìóëà äîïîëíèòåëüíîãî àðãóìåíòà];
5) (t(R cos(sint) > 0 [|sint| ( 1; çíàê êîñèíóñà (åäèíè÷íàÿ îêðóæíîñòü)].
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå; ëîãèêà!):
1) Â.: ñòð. 302, ¹663 (15; 16) [15) ñóììà âçàèìíî îáðàòíûõ; 16) âîçâåäåíèå â êâàäðàò].
Äîêàæèòå, ÷òî:
2) (((R 1 – 4sin((sin(( + [pic]) ( 0 [ôîðìóëû óìíîæåíèÿ].
3) (((R [pic] ( 1. Óñèëüòå íåðàâåíñòâî.
[... = [pic], ãäå m(Z; ( [pic] (åäèíè÷íàÿ îêðóæíîñòü)].
4) [pic] < sin20((sin50((sin70( < [pic] [... = [pic]sin80( è îöåíêà (åäèíè÷íàÿ îêðóæíîñòü)].
5) (x(R sin13x + cos15x ( 1. Ïðè êàêèõ x âûïîëíÿåòñÿ ðàâåíñòâî?
[sin13x ( sin2x; cos15x ( cos2x; ïðè [pic]].
6) (y(R (1 + siny + cosy)(1 – siny + cosy)(1 + siny – cosy)(siny + cosy – 1) ( 1 [Ïåðåìíîæèòü ñêîáêè: ïåðâóþ è ÷åòâåðòóþ; âòîðóþ è òðåòüþ; ... = sin22y].
Äîìàøíåå çàäàíèå: Â.: ñòð. 301 – 302, ¹663 (1; 2; 4; 5). Äîêàæèòå íåðàâåíñòâà: 1) tg1( + tg2( + ... + tg89( > 89; 2) |sin(cosx)| < [pic]; 3) [pic]. Ðåøèòå óðàâíåíèÿ: à) cosx + 6 = x2 + (x
+ (2; á) |tgx| = [pic] – 1.
|Óðîê 37, 38 |17.10. |
|Äîêàçàòåëüñòâî è ðåøåíèå òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Óðàâíåíèÿ: [à) (; á) [pic]].
Ðàññìîòðèì äîêàçàòåëüñòâî íåðàâåíñòâ, ñâÿçàííîå ñ ïðèìåíåíèåì ôîðìóëû äîïîëíèòåëüíîãî àðãóìåíòà, è «ñîïóòñòâóþùèå» çàäàíèÿ.
2. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå; çàïèñè!):
1) Â.: ñòð. 302, ¹663 (10) [ëåâàÿ ÷àñòü: [pic] < 4; [pic]].
2) Â.: ñòð. 302, ¹665 [ðàâíîñèëüíîñòü! |[pic]sin(x + ()| ( [pic]; [pic]]
3) Íàéäèòå îáëàñòü çíà÷åíèé ôóíêöèè: à) y = 5cos2x – 12sinx(cosx; á) f(x) = 2sin([pic] + 4x)(cos(4x – [pic]) [à) y = 6,5cos(2x + t) + 2,5; E(y) = [–4; 9]; á) f(x) = [pic] – cos[pic]; E(f) = [[pic]– 1; [pic] + 1]; î íåïðåðûâíîñòè!].
4) Íàéäèòå ýêñòðåìàëüíûå çíà÷åíèÿ âûðàæåíèÿ: cos2t – 8cost [... = 2(cost – 2)2 – 9 èëè çàìåíà ïåðåìåííîé è èññëåäîâàíèå êâàäðàòè÷íîé ôóíêöèè; –7 è 9].
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì ðåøåíèå ïðîñòåéøèõ òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ, òî åñòü íåðàâåíñòâ âèäà f(t) * m, ãäå f(t) – îäíà èç òðèãîíîìåòðè÷åñêèõ ôóíêöèé; m(R, à * – îäèí èç çíàêîâ ñòðîãîãî èëè íåñòðîãîãî íåðàâåíñòâà. Òàêèå íåðàâåíñòâà íàì çíàêîìû, â ÷àñòíîñòè, ìû ñòàëêèâàëèñü ñ íèìè, èçó÷àÿ ñâîéñòâà òðèãîíîìåòðè÷åñêèõ ôóíêöèé (m = 0). Äëÿ ðåøåíèÿ íåðàâåíñòâ ìîæíî èñïîëüçîâàòü ëèáî ãðàôèêè ôóíêöèé, ëèáî åäèíè÷íóþ îêðóæíîñòü. Ðàññìîòðèì ïðèìåðû (1) è 2) – åäèíè÷íàÿ îêðóæíîñòü, çàòåì – ãðàôèê; 3) – ãðàôèê; 4) – åäèíè÷íàÿ îêðóæíîñòü).
1) sint ( –0,5. Îòâåò: [pic]. 2) cost < [pic]. Îòâåò: [pic].
3) tgt ( [pic]. Îòâåò: [pic]. 4) ctgt > –[pic].Îòâåò: [pic].
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé):
Ðåøèòå íåðàâåíñòâà: à) sint < [pic] [[pic]]; á) cost ( 0,1 [[pic]; ÷åòíîñòü!]; â) tgt > [pic] [[pic]]; ã) ctgt ( 2 [[pic]].
Äîìàøíåå çàäàíèå: Â.: ï. 9 (ñòð. 302 – 305); ¹666 (6). Ðåøèòå íåðàâåíñòâà: à) sint ( –0,9; á) cost > –0,5; â) tgt ( 1. ¹663 (8). Äîêàæèòå íåðàâåíñòâà: 1) |1 + 2[pic]sin((cos( – 2cos2(| ( [pic], ãäå à ( 0; ïðè êàêèõ à è ( – ðàâåíñòâî? 2) [pic]. Íàéäèòå îáëàñòè çíà÷åíèé ôóíêöèé: à) [pic]; á) f(x) = cos2x – sinx. Ïðè êàêèõ çíà÷åíèÿõ c óðàâíåíèå (sin0,5x – 3c + 1)(cosx – c) = 0 èìååò íà [–2(; 2(] íå÷åòíîå êîëè÷åñòâî êîðíåé?
|Óðîê 39 |21.10. |
|Ðåøåíèå òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Îáëàñòè çíà÷åíèé? [à) [1; 4]; á) [–1; 1,25]].
2. Óñòíî: Ðåøèòå íåðàâåíñòâà (åäèíè÷íàÿ îêðóæíîñòü!: 1) cost ( 2; 2) sint ( –1; 3) sint < –[pic]; 4) cost > –1 [1) R; 2) [pic]; 3) (; 4) [pic]].
Ðàññìîòðèì ðåøåíèå áîëåå ñëîæíûõ òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. Äëÿ èõ ðåøåíèÿ èñïîëüçóåòñÿ ÿâíàÿ èëè íåÿâíàÿ çàìåíà ïåðåìåííûõ, ïîýòîìó íåñêîëüêî óäëèíÿåòñÿ çàïèñü ðåøåíèÿ (ïîêàçàòü íà ïðèìåðå ïåðâîãî çàäàíèÿ).
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé): Ðåøèòå íåðàâåíñòâà:
1) sint(cost < [pic] [[pic]];
2) [pic] [[pic]];
3) |2cos((x)| ( 1 [[pic]];
4) [pic] [[pic]].
Äîìàøíåå çàäàíèå: Â.: ñòð. 308, ¹667 (3; 6; 9; 12; 7). Ðåøèòå íåðàâåíñòâà: 1) [pic]; 2) [pic]; 3) |sin(0,5y – 1)| > [pic].
|Óðîê 40, 41 |22.10. |
|Ðåøåíèå òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
Ïðîäîëæèì ðåøåíèå òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. Ðàññìîòðèì áîëåå ñëîæíûå íåðàâåíñòâà, ïðè ðåøåíèè êîòîðûõ î÷åíü âàæíî âûáèðàòü ðàöèîíàëüíûå ñïîñîáû, ïðè÷åì ýòè ñïîñîáû äàëåêî íå âñåãäà ñîâïàäàþò ñî ñïîñîáàìè ðåøåíèé ñîîòâåòñòâóþùèõ òðèãîíîìåòðè÷åñêèõ óðàâíåíèé!
2. Ïèñüìåííî (íà äîñêå è â òåòðàäÿõ, çàòåì – ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Â.: ñòð. 308, ¹667 (2; 4) 2) Ìîæíî ëè ðåøàòü ýòî íåðàâåíñòâî òåì æå ñïîñîáîì, ÷òî è îäíîðîäíîå óðàâíåíèå? Öåëåñîîáðàçíî ëè ðàçëîæåíèå íà ìíîæèòåëè? Ïðè ðåøåíèè òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ òàì, ãäå ýòî âîçìîæíî, ïðèâîäÿò ê îäíîé ôóíêöèè! [[pic]; [pic]]
4) Öåëåñîîáðàçíî ëè çàìåíÿòü äàííîå íåðàâåíñòâî ñîâîêóïíîñòüþ äâóõ ñèñòåì?
[Íåò, ëó÷øå ïîëó÷èòü: [pic]; [pic]]  ýòîì ñëó÷àå, â çàïèñè îòâåòà ìîæíî èñïîëüçîâàòü êàê ðàçíûå áóêâû, òàê è îäèíàêîâûå!
Ðåøèòå íåðàâåíñòâà:
1) sin4t + cos4t(ctg2t ( ( [ctg2t ( (; [pic]].
2) cos2y + 5cosy + 3 ( 0 [2cos2y + 5cosy + 2 ( 0; cosy ( -0,5; [pic]].
3) tgx + 2ctgx ( 3 [tgx = y; [pic]; 0 < tgx ( 1 èëè tgx ( 2; [pic]]
4) 2sin22x + 2cos2x > 3 [Äâà ñïîñîáà: 8cos4x – 10cos2x + 3 < 0; cos2x = t; [pic] èëè 2cos22x – cos2x < 0; 0 < cos2x < 0,5; [pic]]
Òðèãîíîìåòðè÷åñêèå íåðàâåíñòâà èíîãäà âîçíèêàþò ïðè ðåøåíèè èððàöèîíàëüíî - òðèãîíîìåòðè÷åñêèõ óðàâíåíèé èëè êàêèõ - òî äðóãèõ çàäà÷, ñ ÷åì ìû óæå âñòðå÷àëèñü. Êàê ðåøèòü óðàâíåíèå âèäà [pic]? [... ( [pic]]
5) Ðåøèòå óðàâíåíèå: [pic] [[pic]; [pic] (åäèíè÷íàÿ îêðóæíîñòü!)]
6) Íàéäèòå îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé ôóíêöèè [pic] [à) 8sin2x – 2sinx – 1 ( 0; –0,25 ( sinx ( 0,5; D(y) = [pic]; á) f(t) = –8t2 + 2t + 1, t([–0,25; 0,5]; E(y) = [pic]]. Ñëåäóþùèé óðîê – ñ/ð!
Äîìàøíåå çàäàíèå: ïîâòîðèòå îïðåäåëåíèå îáðàòíîé ôóíêöèè, êðèòåðèé îáðàòèìîñòè è ñâîéñòâà âçàèìíî îáðàòíûõ ôóíêöèé (Â.: ñòð. 158 – 160 áåç íåïðåðûâíîñòè èëè òåòðàäü); Â.: ¹667 (1; 5; 10; 13). Ðåøèòå íåðàâåíñòâà: 1) [pic]sint – 2cos2t – 1 ( 0; 2) 2siny(siny – [pic]ctgy) < 3;
3) [pic]; 4) sin3x > 4sin2x. Ðåøèòå óðàâíåíèÿ: à) [pic]; á) [pic].
|Óðîê 42, 43 |24.10. |
|Ðåøåíèå òðèãîíîìåòðè÷åñêèõ íåðàâåíñòâ. Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹4. |
|Ñâîéñòâà âçàèìíî îáðàòíûõ ôóíêöèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû?
2. Íîâûé ìàòåðèàë. Ðàññìîòðèì ïðèìåð ðåøåíèÿ åùå áîëåå ñëîæíîãî òðèãîíîìåòðè÷åñêîãî íåðàâåíñòâà, ðàçîáðàâ íåñêîëüêî ñïîñîáîâ:
cos3x(sinx > 0. Íà ïåðâûé âçãëÿä, íåðàâåíñòâî íå âûãëÿäèò ñëîæíûì, íî
I ñïîñîá. ... ( [pic] èëè [pic]; äâå åäèíè÷íûå îêðóæíîñòè; [pic].
II ñïîñîá. ... ( 0,5(sin4x – sin2x) > 0 ( sin2x(cos2x – 0,5) > 0 ( [pic] èëè [pic] ( ... .  ÷åì ïðåèìóùåñòâî ïî ñðàâíåíèþ ñ I ñïîñîáîì?
III ñïîñîá. ... ( sin4x – sin2x > 0. Ðàññìîòðèì ôóíêöèþ f(x) = sin4x – sin2x, êîòîðàÿ îïðåäåëåíà íà R, íå÷åòíà è ïåðèîäè÷íà. Åå îñíîâíîé ïåðèîä: T = 0,5(2( = (. Íàéäåì x | f(x) = 0, x([0; [pic]]. Ïîëó÷èì: x({0; [pic]; [pic]}. Íàéäåì ïðîìåæóòêè çíàêîïîñòîÿíñòâà íà [pic] (â äàííîì ñëó÷àå – ÷åðåäîâàíèå çíàêîâ!):
[pic]
Îòâåò â íåðàâåíñòâå ïîëó÷èòñÿ, åñëè ê íàéäåííûì ïðîìåæóòêàì äîáàâèòü ÷èñëà, êðàòíûå ïåðèîäó. Ýòîò ñïîñîá – ìåòîä èíòåðâàëîâ â òðèãîíîìåòðèè.
Âû, åñòåñòâåííî, ìîæåòå âûáèðàòü òîò ñïîñîá ðåøåíèÿ, êîòîðûé íðàâèòñÿ.
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå): Ðåøèòå íåðàâåíñòâî:
cos3x ( cos2x [ñîâîêóïíîñòü ñèñòåì èëè ìåòîä èíòåðâàëîâ; ÷åòíîñòü ôóíêöèè! [pic]].
4. Óñòíî (ïî ìàòåðèàëó, ïîâòîðåííîìó äîìà, ñ êðàòêèìè çàïèñÿìè íà äîñêå):
1) Ñôîðìóëèðóéòå Í. è Ä. óñëîâèå îáðàòèìîñòè ôóíêöèè f(x) [(x1(Df, x2(Df x1 ( x2 ( f(x1) ( f(x2)].
2) Ïðèâåäèòå ïðèìåðû ôóíêöèé: à) îáðàòèìîé; á) íåîáðàòèìîé; è îáîñíóéòå.
3) Ñôîðìóëèðóéòå îïðåäåëåíèå ôóíêöèè, îáðàòíîé ê ôóíêöèè f [Ôóíêöèÿ f–1 íàçûâàåòñÿ îáðàòíîé ê ôóíêöèè f, åñëè (x(Df | f(x) = y âåðíî, ÷òî f–1(y) = x].
4) Ïåðå÷èñëèòå ñâîéñòâà âçàèìíî îáðàòíûõ ôóíêöèé [à) (x(Df f–1(f(x)) = x è (y(Ef f(f–1(y)) = y; á) Df = Ef–1 è Ef = Df–1; â) ãðàôèêè ñèììåòðè÷íû îòíîñèòåëüíî ïðÿìîé y = x; ã) îáðàòíàÿ ê ìîíîòîííîé – ìîíîòîííàÿ òîãî æå âèäà]
5) Êàêèå èç îáùèõ ñâîéñòâ ôóíêöèè äîñòàòî÷íû äëÿ åå îáðàòèìîñòè? [Ñòðîãàÿ ìîíîòîííîñòü]
6) Êàêèå èç îáùèõ ñâîéñòâ ôóíêöèè ïðîòèâîðå÷àò îáðàòèìîñòè? [×åòíîñòü èëè ïåðèîäè÷íîñòü]
7) ßâëÿþòñÿ ëè îáðàòèìûìè òðèãîíîìåòðè÷åñêèå ôóíêöèè?
8) Êàê ñäåëàòü èõ îáðàòèìûìè? [Ðàññìîòðåòü íà êàêîì - íèáóäü èç ïðîìåæóòêîâ ìîíîòîííîñòè]
Ýòèì ìû çàéìåìñÿ íà ñëåäóþùåì óðîêå.
5. Ñàìîñòîÿòåëüíàÿ ðàáîòà ¹3 (30 ìèíóò).
Îòâåòû.
|I âàðèàíò. |II âàðèàíò. |
|¹1. ë. ÷.: 2 + 2,5sin(2( + arctg[pic]) ( 4,5. |¹1. ë. ÷.: 1,5 – 2,5sin(2( + arctg[pic]) ( –1. |
|¹2. à) [pic]; |¹2. à) [pic]; |
|á) [pic]. |á) [pic]. |
|¹3. D(y) = [pic]. E(y) = [0; 1]. |
Äîìàøíåå çàäàíèå: Â.: ï. 10 (ñòð. 305 – 307); ¹659 (13); ¹667 (8); ðåøèòå óðàâíåíèå [pic]; ðåøèòå íåðàâåíñòâà: 1) sin3x > sin5x; 2) 4 + sinx + [pic]cosx ( 4cos2(x + [pic]); äîêàæèòå, ÷òî (x ( [pic], n(Z è b ( 0 [pic]. Äîï. ê ñ/ð.
|Óðîê 44 |28.10. |
|Îáðàòíûå òðèãîíîìåòðè÷åñêèå ôóíêöèè, èõ ñâîéñòâà è ãðàôèêè. |
1. Ðàçáîð c/ð.
2. Ïðîâåðêà ä/ç: âîïðîñû? Äîêàçàòåëüñòâî íåðàâåíñòâà? [ë. ÷.: 0,5|sin2x| ( 0,5; ïð. ÷.: ( 0,5]. Ðåøåíèå íåðàâåíñòâ? [1) [pic]2) [2sin(x + [pic])(sin(x + [pic]) – 0,5) ( 0; [pic]].
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì êàæäóþ èç òðèãîíîìåòðè÷åñêèõ ôóíêöèé íà îäíîì èç ïðîìåæóòêîâ ñòðîãîé ìîíîòîííîñòè. Òîãäà äëÿ êàæäîé èç íèõ ñóùåñòâóåò îáðàòíàÿ ôóíêöèÿ. Ïîñòðîèì ãðàôèêè ôóíêöèé íà âûáðàííûõ ïðîìåæóòêàõ, çàòåì ãðàôèêè ôóíêöèé, èì îáðàòíûõ, è âûïèøåì íàçâàíèÿ è ñâîéñòâà ïîëó÷åííûõ ôóíêöèé, èñïîëüçóÿ ââåäåííûå ðàíåå îïðåäåëåíèÿ è ñâîéñòâà âçàèìíî îáðàòíûõ ôóíêöèé:
|y = sinx, [pic] |y = cosx, [pic] |y = tgx, [pic] |y = ctgx, [pic] |
|ãðàôèê |ãðàôèê |ãðàôèê |ãðàôèê |
|y = arcsinx |y = arccosx |y = arctgx |y = arcctgx |
|D(y) = [–1; 1] |D(y) = [–1; 1] |D(y) = R |D(y) = R |
|E(y) = [pic] |E(y) = [0; (] |E(y) = [pic] |E(y) = (0; () |
|âîçðàñòàþùàÿ |óáûâàþùàÿ |âîçðàñòàþùàÿ |óáûâàþùàÿ |
|íå÷åòíàÿ |arccos(–x) = ( – arccosx |íå÷åòíàÿ |arcctg(–x) = ( – arcctgx |
Ïîñëåäíåå ñâîéñòâî äëÿ ôóíêöèé cosx è ctgx ïîçæå áóäåò äîêàçàíî ñòðîãî!
4. Óïðàæíåíèÿ (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå èëè óñòíî):
1) Íàéäèòå îáëàñòè îïðåäåëåíèÿ ôóíêöèé:
à) [pic] [[pic] ( x([1; 5]]; á) y = arccos(4x2) [D(y) = [–0,5; 0,5]]; â) y = arctg([pic]) [D(y) = [0; +()]; ã) y = arcctg(x3 – x) [D(y) = R]; ä) y = arcsin(cosx) [D(y) = R].
2) Äëÿ ïóíêòîâ â) è ä) óêàæèòå ìíîæåñòâî çíà÷åíèé ôóíêöèè [â) [0; [pic]); ä) [pic]].
3) Ïîñòðîéòå ãðàôèê ôóíêöèè y = 2arcsin(0,5x –2) – 1.
Äîìàøíåå çàäàíèå: Â.: ï. 1 (ñòð. 310 – 312); ¹670 (1; 2); 1) ïîñòðîéòå ãðàôèêè óðàâíåíèé: à) |y| = arcsinx; á) y = arccos|x|; â) |y| = |arctgx|; ã) |y| = arcctg|x|; 2) ðåøèòå íåðàâåíñòâà: à) 2sin2(x + [pic]) – 2 ( sinx – cosx; á) [pic]; â) 4sinx(sin2x(sin3x > sin4x.
|Óðîê 45, 46 |29.10. |
|Òîæäåñòâà äëÿ îáðàòíûõ òðèãîíîìåòðè÷åñêèõ ôóíêöèé. |
1. Ïðîâåðêà ä/ç: âîïðîñû ïî ãðàôèêàì? Îòâåòû â íåðàâåíñòâàõ? [à) [pic]; á) [pic]]
2. Óñòíî: 1) Äëÿ êàêèõ x ñïðàâåäëèâû ðàâåíñòâà è ïî÷åìó: à) [pic]; á) [pic]?
[Ïðè ÷åòíûõ n äëÿ x ( 0; ïðè íå÷åòíûõ n – äëÿ ëþáûõ; à) [pic]; á) [pic]].
2) Ìîãóò ëè âûðàæåíèÿ â ëåâîé ÷àñòè ïðèíèìàòü äðóãèå çíà÷åíèÿ?
[à) íåò; á) äà, , åñëè n – ÷åòíîå, x < 0, òî [pic]].
3) Êàêîå ñâîéñòâî âçàèìíî îáðàòíûõ ôóíêöèé îòðàæàþò ðàññìîòðåííûå ðàâåíñòâà?
[(x(Df f–1(f(x)) = x è (y(Ef f(f–1(y)) = y]
3. Íîâûé ìàòåðèàë. Ðàññìîòðèì ïðèìåíåíèå ýòîãî ñâîéñòâà â òðèãîíîìåòðèè.
Âû÷èñëèòå è îáîñíóéòå:
1) à) sin(arcsin[pic]) [[pic]]; á) sin(arcsin[pic]) [[pic]]; â) sin(arcsin1,5) [íå ñóùåñòâóåò].
Âûâîä: (x([–1; 1] sin(arcsinx) = x.
2) à) cos(arccos[pic]) [–0,5]; á) cos(arccos0,3) [0,3]; â) cos(arccos(–2)) [íå ñóùåñòâóåò].
Âûâîä: (x([–1; 1] cos(arccosx) = x.
3) à) tg(arctg[pic]) [[pic]]; á) tg(arctg(–7) [–7].
Âûâîä: (x(R tg(arctgx) = x.
4) à) ctg(arcctg(–1)) [–1]; á) ctg(arcctg(() [(]. Âûâîä: (x(R ctg(arcctgx) = x.
4. Ïèñüìåííî (íà äîñêå è â òåòðàäÿõ, çàòåì – ñàìîñòîÿòåëüíî ñ ïðîâåðêîé íà äîñêå):
Äîêàæèòå òîæäåñòâà: 1) (x([–1; 1] arcsinx + arccosx = [pic] [... ( arcsinx = [pic] – arccosx; sin(arcsinx) = x è sin([pic] – arccosx) = cos(arccosx) = x, ïðè÷åì, [pic] ( arcsinx ( [pic] è 0 ( arccosx ( ( ( –( ( –arccosx ( 0 ( [pic] ( [pic] – arccosx ( [pic]]. Ýòî òîæäåñòâî ïîëåçíî ïîìíèòü!
2) (x(R arcctg(–x) = ( – arcctgx [ctg(arcctg(–x)) = –x è ctg(( – arcctgx) = ctg(– arcctgx) = –ctg(arcctgx) = –x, ïðè÷åì, 0 ( arcctg(–x) ( ( è 0 ( arcctgx ( ( ( –( ( –arcctgx ( 0 ( 0 ( ( –arcctgx ( (].
5. Íîâûé ìàòåðèàë. Âû÷èñëèòå è îáîñíóéòå:
1) à) [pic] [[pic]]; á) [pic] [[pic], òàê êàê [pic]]; â) arcsin(sin1,5) [1,5]. Âûâîä: [pic] arcsin(sinx) = x.
2) à) [pic] [[pic]]; á) [pic] [[pic], òàê êàê [pic]]; â) arccos(cos3) [3]. Âûâîä: [pic] arccos(cosx) = x.
3) à) [pic] [[pic]]; á) [pic] [[pic], òàê êàê [pic]]; â) arctg(tg[pic]) [[pic]]. Âûâîä: [pic] arctg(tgx) = x.
[pic]
[pic]
4) à) [pic] [[pic]]; á) [pic] [[pic], òàê êàê [pic]]; â) arcctg(ctg[pic]) [[pic]]. Âûâîä: [pic] arcctg(ctgx) = x.
Ðèñ. 1à
Ðèñ. 1á
Îáðàòèòå âíèìàíèå, ÷òî äëÿ äðóãèõ x âûðàæåíèÿ â ëåâîé ÷àñòè èìåþò ñìûñë, íî íå ðàâíû x!
[pic]
[pic]
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðèñ. 1ã
Ðèñ. 1â
1) Âû÷èñëèòå: à) arcsin(sin200() [[pic]]; á) arcctg(ctg[pic]) [[pic]]; â) arccos(sin[pic] [[pic]]; ã) arccos(cos(–4)) [2( – 4]; ä) arcsin(cos5) [5 – 1,5(].
2) Ïîñòðîéòå ãðàôèêè ôóíêöèé: à) y = arcsin(sinx); á) y = arccos(cosx); â) y = arctg(tgx); ã) y = arcctg(ctgx) [Ñì. ðèñ. 1 à – ã].
Äîìàøíåå çàäàíèå: òîæäåñòâà – çíàòü; ¹640; ¹647; ¹675 (1) – çíàòü; äîêàæèòå, ÷òî (x([–1; 1] arccos(–x) = ( – arccosx; ïîñòðîéòå ãðàôèêè ôóíêöèé (ïî âàðèàíòàì: I – à) è ã); II – á) è â)): à) y = sin(arcsinx); á) y = cos(arccosx); â) y = tg(arctgx); ã) y = ctg(arcctgx). Íàéäèòå ýêñòðåìàëüíûå çíà÷åíèÿ ôóíêöèè: y = arcsin3x + arccos3x.
|Óðîê 47, 48 |31.10. |
|Ïðèìåíåíèå ñâîéñòâ îáðàòíûõ òðèãîíîìåòðè÷åñêèõ ôóíêöèé |
|äëÿ âû÷èñëåíèé è äîêàçàòåëüñòâ. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Ãðàôèêè – ïðîâåðèòü [à), á) y = x, x([–1; 1]; â), ã) y = x]
2. Íîâûé ìàòåðèàë. Ðàññìîòðèì âû÷èñëèòåëüíûå çàäàíèÿ, ñâÿçàííûå ñ îáðàòíûìè òðèãîíîìåòðè÷åñêèìè ôóíêöèÿìè.
Ïðèìåð. Âû÷èñëèòå: ctg(arccos(–0,8)).
[pic]
Ïóñòü arccos(–0,8) = (, òîãäà cos( = –0,8; (([0; (]. Íàéäåì ctg(: 1) |sin(| = 0,6; 2) òàê êàê (([0; (], òî sin( > 0; ctg( = [pic]. Îòâåò: ctg(arccos(–0,8)) = [pic].
3. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ óñòíîé ïðîâåðêîé):
1) Âû÷èñëèòå: à) sin(arctg[pic]) [0,6]; á) tg(2arcsin[pic]) [[pic]]; â) arctg1 + arctg2 + arctg3 [arctg2 = (; arctg3 = (; tg(( + () = –1 è 0 < ( + ( < (; ( + ( = [pic]; Îòâåò: (].
 ïóíêòå â) ïîñòðîéòå ãåîìåòðè÷åñêóþ èíòåðïðåòàöèþ ïîëó÷åííîãî ðàâåíñòâà è äîêàæèòå åãî ìåòîäàìè ãåîìåòðèè [... = (DOA + (DOB +(DOC = 45( + (90( – (DBO) + (90( – (DCO) = 225( – (BOE = 180(, òàê êàê (BOE = (BEO = 45( ((EBO – ðàâíîáåäðåííûé ïðÿìîóãîëüíûé; ñì. ðèñ.].
2) Äîêàæèòå: arcsin[pic] + arccos[pic] = arcsin[pic] [arcsin[pic] = (; arccos[pic] = (; sin(( + () = [pic] è 0 < ( + ( < [pic]]
4. Íîâûé ìàòåðèàë. Ðàññìîòðèì äîêàçàòåëüñòâî òîæäåñòâ, ñîäåðæàùèõ ïåðåìåííûå.
Ïðèìåð. Â.: ñòð. 318, ¹675 (2).
Ïóñòü arcsinx = ( ( sin( = x è [pic]; arccos[pic] = ( ( cos( = [pic] è (([0; (]; |cos(| = cos( = [pic] = cos(. Åñëè 0 ( x ( 1, òî [pic], òî åñòü, ( = (. Åñëè –1 ( x ( 0, òî [pic], òî åñòü, ( = –(.
5. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Â.: ñòð. 318, ¹675 (4, 7, 8) [Àíàëîãè÷íî 2)]
Äîìàøíåå çàäàíèå: Â.: ¹650 (5; 7); ¹675 (3; 5 – îïå÷àòêà: â ëþáîé èç ÷àñòåé arcctg; 6). 1) Âû÷èñëèòå: [pic]. 2) Íàéäèòå îáëàñòü îïðåäåëåíèÿ ôóíêöèè y = arccos([pic]ctg((x). 3) Ïîñòðîéòå ãðàôèêè ôóíêöèé (ïî âàðèàíòàì: I – à) è ã); II – á) è â)): à) y = tg(arctg(0,5x3)); á) y = ctg(arcctg(0,5x3)); â) y = sin(arcsin(x2 – 1)); ã) y = cos(arccos(x2 – 1)).
Äîêàæèòå, ÷òî [pic].
|Óðîê 49, 50 |11.11. |
|Ðåøåíèå óðàâíåíèé è íåðàâåíñòâ, ñîäåðæàùèõ îáðàòíûå òðèãîíîìåòðè÷åñêèå ôóíêöèè. |
1. Ïðîâåðêà ä/ç: âîïðîñû? Ãðàôèêè – çàãîòîâèòü íà äîñêå [à), á) y =0,5x3; â), ã) y = x2 – 1, –1 ( x2 – 1 ( 1 ( |x| ( [pic]].
2. Óñòíî: 1) Âû÷èñëèòå: à) [pic] [[pic]]; á) [pic] [[pic]]; â) [pic] [íå ñóùåñòâóåò]; ã) [pic] [[pic]]; ä) tg(arctg2,1() [2,1(]; å) arctg(tg2,1() [0,1(]; æ) ctg(arcctg(2) [(2]; ç) arcctg(ctg(2) [(2 – 3(]. 2) Â.: ñòð. 318, ¹675 (9; 10) – íå äîêàçûâàÿ, âñïîìíèòå, ãäå ìû âñòðå÷àëèñü ñ èíòåðïðåòàöèåé ýòèõ òîæäåñòâ? [Ïðè ïîñòðîåíèè ãðàôèêîâ ôóíêöèé, çàïèñàííûõ â ëåâîé ÷àñòè]
3. Íîâûé ìàòåðèàë. ×òî ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ [pic] = a, ãäå à(R è ïî÷åìó?
[Ïðè à < 0 (, òàê êàê à(E([pic]); ïðè à ( 0 x = a2 (ïî îïðåäåëåíèþ àðèôìåòè÷åñêîãî êâàäðàòíîãî êîðíÿ)].
Îáîáùèòå ïîëó÷åííûé ðåçóëüòàò [f–1(x) = a ( x = f(a), åñëè à(E(f–1) è f–1(x) = a ( x((, à(E(f–1)].
Ðàññìîòðèì ïðèìåðû óðàâíåíèé, ñîäåðæàùèõ îáðàòíûå òðèãîíîìåòðè÷åñêèå ôóíêöèè. Ïðèìåðû. 1) arctgx = 0,5(2 ( x((, òàê êàê 0,5(2([pic]; 2) arccosx = 3 ( x = cos3, òàê êàê 3([0; (].
4. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðåøèòå óðàâíåíèÿ: 1) arccos(x2 – 4x + 2) = ( [1; 3];
2) arcsinx(arccosx = [pic] [òîæäåñòâî; arcsinx = [pic] èëè arcsinx = [pic]; 0,5; [pic]];
3) arcsin2y = arccosy [... ( sin(arcsin2y) = sin(arccosy) ( 2y = [pic]; [pic]];
4) arctgx + arctg[pic] + arctg[pic] = [pic] [1. Ìîíîòîííîñòü!].
5. Íîâûé ìàòåðèàë. ×òî ÿâëÿåòñÿ ðåøåíèåì íåðàâåíñòâà [pic] < a, ãäå à > 0 è ïî÷åìó?
[0 ( x < a2; îáëàñòü îïðåäåëåíèÿ è ìîíîòîííîñòü] Ýòè æå ñâîéñòâà ôóíêöèé ïðèìåíÿþòñÿ è ïðè ðåøåíèè íåðàâåíñòâ, ñîäåðæàùèõ îáðàòíûå òðèãîíîìåòðè÷åñêèå ôóíêöèè.
Ïðèìåðû. 1) arcctgx < [pic] ( arcctgx < arcctg[pic] ( x > [pic] (ïðè íåîáõîäèìîñòè – ãðàôèê!). Îòâåò: ([pic]; +().
2) arcsinx ( [pic] ( arcsinx ( arcsin(sin[pic]) ( [pic] ( sin[pic] ( x ( 1. Îòâåò: [sin[pic]; 1].
6. Ïèñüìåííî (ñàìîñòîÿòåëüíî â òåòðàäÿõ ñ ïðîâåðêîé íà äîñêå):
Ðåøèòå íåðàâåíñòâà (ðàöèîíàëüíîñòü!): 1) arcsin[pic] ( 2arccos[pic]
[[pic] = t; arcsint ( [pic] ( [pic] [pic] ( [pic] ( 1 ( 1 ( x ( [pic]. Îòâåò: [pic]];
2) [pic] [[pic]; [pic] ( [pic] èëè [pic]; ïåðâîå íåðàâåíñòâî ðåøåíèé íå èìååò; –1 ( 1 – x2 ( –0,5 ( [pic]. Îòâåò: [pic]];
3) Â.: ñòð. 319, ¹678 (2)
[arctg2x = t; 2 ( |t – 3| ( 4; –1 ( arctg2x ( 1 èëè 5 ( arctg2x ( 7; âòîðîå íåðàâåíñòâî ðåøåíèé íå èìååò; –tg1 ( 2x ( tg1. Îòâåò: [pic]].
Äîìàøíåå çàäàíèå: Â.: ï. 5 (ñòð. 318 – 319); ¹676 (2; 3); ¹677 (1); ¹678 (1). Ðåøèòå óðàâíåíèÿ èëè íåðàâåíñòâà: 1) 5arctgt + 3arcctgt = 2(; 2) arcsin|x| > arccos|x|; 3) arcsin(3 – 2x) + arccos(x2 – 1) = (.
Äëÿ ñàìîïðîâåðêè: [1) òîæäåñòâî; arctgt = [pic]; Îòâåò: 1. 2) [|x| = y; arcsiny > [pic] ( [pic] [pic]. Îòâåò: [pic]; 3) Îòâåò: 1. Ìîíîòîííîñòü!].
|Óðîê 51, 52 |14.11. |
|Êîíòðîëüíàÿ ðàáîòà ¹3. Çà÷åò ¹1 ïî òåìå «Òðèãîíîìåòðèÿ. |
1. Êîíòðîëüíàÿ ðàáîòà ¹3 (45 ìèíóò).
Îòâåòû.
|I âàðèàíò. |II âàðèàíò. |
|¹1. à) 0,28; á) 6 – (. |¹1. à) –0,28; á) 4 – (. |
|¹2. y = 2x – 1, 0 ( x ( 1. |¹2. y = 0,5x + 1, –4 ( x ( 0. |
|¹3. à) [pic]; |¹3. à) [pic]; |
|á) [pic]. |á) [pic]. |
|¹4. à) y = [pic] – arccos(cosx); á) y = [pic]. |¹4. à) y = [pic] – arcsin(sinx); á) y = [pic]. |
2. Çà÷åò ¹1. 15 áèëåòîâ ïî 4 âîïðîñà.
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