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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