Seminar 7: de nirea functiilor trigonometrice tg si cotg ...
[Pages:1]Seminar 7: denirea functiilor trigonometrice tg si cotg, proprietatile lor, formule trigonometrice
1. Demonstrati urmatoarele formule trigonometrice, pentru acele valori ale lui x, y, etc. pentru care expresiile
au sens.
sin2
x
=
1
tg2x + tg2x ;
cos2
x
=
1
1 + tg2x ;
tgx ? tgy
2tgx
ctgxctgy 1
tg (x ? y) =
;
1 tagxtgy
tg2x = 1 - tg2x ;
ctg (x ? y) =
; ctgx ? ctgy
x sin x tg =
2 1 + cos x
x 1 - cos x ; tg =
2 sin x
cos x
=
1 1
- +
tg2
x 2
tg2
x 2
;
sin x
=
1
2tg
x 2
+
tg2
x 2
;
sin (x ? y)
sin (x - y)
tgx + tgy
ctgx + ctgy
tgx ? tgy =
; ctgx - ctgy = -
; tgxtgy =
; ctgxctgy =
.
cos x cos y
sin x sin y
ctgx + ctgy
tgx + tgy
2.
Daca
a
(0,
2
),
b
(
3 2
,
2)
si
sin a =
1 2
,
cos b =
3 2
,
calculati
tg(a
+
b).
3.
Stiind
ca tgx =
m n
,
m,
n
Z,
n
=
0,
calculati
E = m sin 2x + n cos 2x;
4.
Daca 5 cos x + 10 sin x - 11 = 0, calculati
cos x, sin x,
tg
x 2
.
5. Vericati identitatile.
(a)
cos(a+b) cos a cos b
=
1 - tgatgb;
(b)
; sin(a+b)
sin(a-b)
=
1+ctgactgb 1-ctgactgb
(c)
cos 3x sin x
+
sin 3x cos x
=
2ctg2x;
(d) tg3a - tg2a - tga = tg3atg2atga;
(e)
tg(a + b + c) =
; tga+tgb+tgc-tgatgbtgc
1-tgatgb-tgatgc-tgbtgc
(f )
; sin x+sin 3x+sin 5x+sin 7x+sin 9x cos x+cos 3x+cos 5x+cos 9x
=
tg5x
(g)
1
+
cos
a
+
ctg
a 2
=
2ctg
a 2
sin
a 2
+
4
cos
a 2
-
4
.
1
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