Уравнения, однородные относительно sin x и cos x

Уравнения, однородные относительно sin x и cos x

620(2) 620(2), 621(2,4), 622(3,4), 644(1) 1 1 cos x cos x , x arccos 1 2k , 2 2 2 k Z; 1 cos 2 x 1 1 1 x arccos 2k ; ; cos x ; 2 2

2 2 3 1 cos 2 x 1 ; x 4 2k , cos 2 x 0 ; k Z; x 2k ; 2 x n , n Z ; 4 2 n n x ,n Z ; : ,n Z . 4 2 4 2 2 621(2, 4) 620(2), 621(2,4), 622(3,4), 644(1)

2 )3 cos 2 x sin x 1 0 ; sin x 1 , 3 1 sin 2 x sin x 1 0 ; 2 sin x ; 3 sin 2 x sin x 2 0 ; 3 sin x ; 1 1 ; 2 3 2 0 ; x 2 2n , n Z , D 1 24 25 ; 1 5 1 , ; 6 2 ; 3

x 1 k arcsin 2 k , k Z ; 3 2 k : 2n , n Z ; 1 arcsin k , k Z . 2 3 621(2, 4) 620(2), 621(2,4), 622(3,4), 644(1) 4 )2 sin 2 x 3 cos x 0 ; 2 1 cos 2 x 3 cos x 0 ; 2 cos 2 x 3 cos x 2 0 ; osxosx ; 1 1 ; 2 2 3 2 0 ; D 9 16 25 ; 1 3 5 , ; 2 4

2 ; 1 1 ; 1 cos x ; 2 2 2n , n Z 3 2 2 : 2n , n Z ; 2n , n Z . 3 3 622(3, 4) 3 )tg 2 x 3 tgx 4 0 ; tgx ; y 2 3 y 4 0 ; y 1 , y 4 ;

tgx 1 , tgx 4 ; x 4 n , n Z x arctg 4 n ; : n , n Z , arctg 4 n , n Z . 4 620(2), 621(2,4), 622(3,4), 644(1) 4 )tg 2 x tgx 1 0 ; tgx ; y 2 y 1 0 ; D 1 4 0 ; ; : . 644(1) 620(2), 621(2,4), 622(3,4), 644(1) 4 cos x 3 4 sin 2 x ; 2 2 2 ; 2 4 cos x 3 4 1 cos x ;

4 1 2 4 cos 2 x 4 cos x 1 0 ; , 2 cos x 0 , 1 ) 1 2 2 4 cos x 4 cos x 1 0 ; 2 ; 1 1; cos x y , 1 1 ; 1 2 cos x , cos x 0 2 4 4 1 0 ; 2 D 1 2 4 4 8 ; x arccos 2k ; k Z ; 4 2

2 644(1) 4 cos x 3 4 sin x ; 620(2), 621(2,4), 622(3,4), 644(1) 2 4 cos x 4 cos x 1 0 ; 1 2 , 1 1; 2 cos x 0 , 2 ) 2 1 2 4 cos x 4 cos x 1 0 ;

; 2 cos x y , 1 1 ; 1 2 4 2 4 1 0 ; cos x , cos x < 0 2 D 4 4 8 ; 1 2 x arccos 2k ; k Z ; 4 2 2 2 2 ; 1 2 2 1 4 : arccos 2k ; arccos 2k , k Z . 2 2

02/08/20 , sin x cos x 110 14 : : 1) a sin x + b cos x = 0 ( ); 2) asin2x+ bsin x cos x + ccos2x = 0 ( ). a sin x + b cos x = 0 : asin2x+ bsin x cos x + ccos2x = 0 cos x (cos2x ) , : a tg x + b = 0 (a tg2x + b tg x + c = 0). , , . , cos x = 0 cos2x = 0 .

a sin x + b cos x = 0 asin2x+ bsin x cos x + ccos2x = 0 cos x = 0, asin x + b0 = 0, . . sin x = 0. , cos2x = 0, sin2x=0. , sin x cos x , sin2x + cos2x = 1. , cos x ( cos2x) . 624(1, 3) 636(1, 3) a sin x + b cos x = 0 asin2x+ bsin x cos x + ccos2x = 0 : 1) a sin x + b cos x = , 0, b 0, c 0 ( ); 2) asin2x+ bsin x cos x + ccos2x = d, d 0 ( ).

a sin x + b cos x = 1) . x 2 x cos x cos sin 2 2 2 x x sin x 2 cos sin 2 2 x 2 x 1 cos sin 2 2 2 x x 2 x 2 x

2 x 2 x a 2 cos sin b cos sin c cos sin ; 2 2 2 2 2 2 x x 2 x 2 x b c sin 2a cos sin b c cos 0 ; 2 2 2 2 . a sin x + b cos x = 2) . 2 x x 1 tg

2 x 2tg 1 tg 2 cos x 2 b 2 c ; a 2 x 1 tg 2 x 2 x 2 1 tg 1 tg 2 2 x x 2tg 2 x 2 x 2a tg b b tg c c tg ; 2 sin x 2 2 2 2 x x

1 tg 2 x b osx tg 2a tg b c 0 ; 2 2 . 2 a sin x + b cos x = 3) . a 0 , b 0 , osx a 2 b 2 , a 2 a b 2 sin x osx sin b 2 a b

b a 2 b2 2 cos x , cos c a 2 b2 a a 2 b2 . , sin2 + cos2 = 1 : sin x a sin x + b cos x = cos sin x sin cos x c

c a 2 b2 a 2 b2 osx arcsin b 2 a b 2 arccos a a 2 b2 b osx arctg , osx a , osx . : 625(1) 625(3) asinx + bcosx = c |a| = 1a|a| = 1 = 1 |a| = 1b|a| = 1 = 1, :

sin x cos x 2 sin x ; sin x cos x 2 cos x 4 4 asin2x+ bsin x cos x + ccos2x = d sin2x + cos2x d. asin2x+ bsin x cos x + ccos2x = d(sin2x + cos2x) . : 623(2) 647 647 sin2x - sin x cos x 2 cos2x = sin2x - sin x cos x 2 cos2x = (sin2x + cos2x) (1 )sin2x - sin x cos x (2 + a) cos2x = 0 cos2x: (1 )tg2x - tg x (2 + a) = 0 tg x = y, : (1 ) 2 (2 + ) = 0 D= 1 + 4(1 )(2 + a) = 1 + 8 8a + 4a 4a2 = = - 4a2 4a + 9 , D < 0. - 4a2 4a + 9 < 0 - 4a2 4a + 9 = 0 D/4 = 4 + 36 = 40

4a2 + 4a - 9 = 0 647 - 4a2 4a + 9 < 0 D = 4 + 36 = 40 1 10 , 2 1 10 ; 2 - //////////// + 1 10 2 \\\\\\\\\\\\ 2 40

, 4 2 40 ; 4 1 10 2 a 1 10 1 10 a a . 2 2 1 10 1 10 : a a . 2 2 :

sin x cos x 2 cos 15 x 6 cos x 2 sin 2 6 2 2 2 6 2 2 cos x sin 2 2 2 2 2 2 3 1 1 cos x sin 2 2 2 1 cos cos x sin sin 6 6

2 1 arccos 2 k , k Z ; 6 2 1 cos x ; 6 2 6 cos x 2 sin 2 1 arccos 2 k , k Z ; 6 2 6 4 2 k , k Z , 2 k , k Z ; 6 4

5 12 2k , k Z , 2k , k Z ; 12 5 19 19 k 1 , x 2 ; ; 12 12 12 12 k 0 , x ; 12 : . 12

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