МАТЕМАТИЧНИЙ ГУРТОК - uCoz

МАТЕМАТИЧНИЙ ГУРТОК - uCoz

, 1. . 2. . 3. . 4. . 5. .

6. . : 2 2 sin x sin x 1 0 : sin x t 2 2t t 1 0 t1 1, t 2 0,5 sin x 1 x 2n, n Z 2 k

sin x 0,5 x ( 1) k , k Z 6 : 2 6 sin x 5 cos x 2 0 2 6 1 cos x 5 cos x 2 0 2

6 cos x 5 cos x 4 0 : cos x t 2 6t 5t 4 0 t1 0,5; t 2 4 / 3 2 cos x 0,5 x 2n, n Z 3 cos x 4 / 3 1 ' : tgx 2ctgx 3 k x n, n Z :

, x ,k Z 2 2 x m, m Z 2 2 tgx 3 tg x 3tgx 2 0 tgx 2 : tgx t t 3t 2 0 t1 1 t 2 2 tgx 1 x n, n Z 4

tgx 2 x arctg 2 k , k Z : 2 cos 2 x 4 cos x cos x 0 2 2 2 cos x 1 4 cos x cos x 0 2 6 cos x cos x 1 0 2 : cos x t 6t t 1 0

t1 0,5 t 2 2 / 3 2 cos x 0,5 x 2n, n Z 3 cos x 2 / 3 x arccos(2 / 3) 2k , k Z sinx+bcosx=0 (a0, b0) . 3 sin x cos x 0 2 sin x 3 cos x 0 sin x cos x

1. asinx+bcosx=0 cosx0. 2. atgx+b=0. 3. (cosx0)). : 3 sin x cos x 0 1 3tgx 1 0 tgx 3 x n, n Z 6

: x n, n Z 6 sin2x+bsinxcosx+ccos2x=0 (a0, b0, c0 0) . 2 2 2 sin x 5 sin x cos x 3 cos x 0 2 4 sin x sin 2 x 0 2

2 3 cos x 5 sin x sin 2 x 1 1. asin2x+bsinxcosx+ccos2x=0 cos2x0. 2. atg2x+btgx+c=0. 3. (cosx0)). : 2

2 2 sin x 5 sin x cos x 3 cos x 0 2 2tg x 5tgx 3 0 : tgx t 2 2t 5t 3 0 t1 1, t 2 1,5 tgx 1, tgx 1,5 x n, n Z 4 x arctg1,5 k , k Z :

2 4 sin x sin 2 x 3 2 2 2 2 sin x 2 sin x cos x 3(sin x cos x) 2 2 sin x 2 sin x cos x 3 cos x 0 2 tg x 2tgx 3 0 : tgx t

2 t 2t 3 0 t1 1, t 2 3 tgx 1, x n, n Z 4 tgx 3 x arctg 3 k , k Z : sin x sin 2 x sin 3x sin 4 x 0 (sin x sin 3 x) (sin 2 x sin 4 x) 0 2 sin 2 x cos x 2 sin 3 x cos x 0

2 cos x(sin 2 x sin 3 x) 0 5x x 2 cos x 2 sin cos 0 2 2 cos x 0 x n, n Z 2 x x sin 0 k , x 2k , k Z 2 2 5x 5x

cos 0 m, 2 2 2 2m x ,mZ 5 5 2n : n, 2n, ,nZ 2 5

5 : 1 cos x cos 2 x 0 2 2 cos x cos x 0 cos x(2 cos x 1) 0 cos x 0 2 cos x 1 0 1 cos x 2 cos x 0 x n, n Z 2 1

2 cos x x 2k , k Z 2 3 2 : n, 2 , n Z 2 3 : cos 3 x sin 5 x 0 cos 3x cos 5 x 0 2

2 cos x cos 4 x 0 4 4 cos 4 x 0 cos 4 x 0

4 x n, x n, 4 2 4 2 3 x n, n Z 4 4 x k , 4 x k , 4 2 4 2 3 k x ,k Z

16 4 3 3 : n, ,nZ 4 16 4 acosx+bsinx=c, a0,b0,c0 1.

acosx+bsinx=c a +b . 2 a : 2 a b a 2 2

2. a b 2 cos x b 2 a b 2

b 2 2 a b 2 1 cos cos x sin sin x

arccos a2 b2 , sin a2 b2 c 2 c cos c

3. : 4. x (1) (2). sin x 2 a b 2 2 a 2

a b b 2 a b cos( x ) 2n, n Z 2 2 (1) ( 2) c

a2 b2 : cos x 3 sin x 2 . . a 1, b 3 , 2 2 1 ( 3 ) 2 1 3 cos x sin x 1

2 2 cos cos x sin sin x 1 3 3 cos x 1 x 2n, n Z 3 3 x 2n, n Z 3

: 2n, n Z 3 1. (x=+2n,nZ)). 2. x 2tg 2 sin x 2 x

1 tg 2 x 1 tg 2 cos x 2 x 1 tg 2 2 x 2tg 2 tgx 2 x 1 tg

2 3. : 4. , x=+2n,nZ) . 5. : 2 sin x cos x 1 x 2 x 2tg 1 tg 2 2 3

1 2 x 2 x 1 tg 1 tg 2 2 x 2 x 2 x 6tg 1 tg 1 tg 2 2 2 x 1 1

tg x 2arctg 2n, n Z 2 3 3 : 3 sin cos 1 1 : 2n, 2arctg 2n, n Z 3

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