Question

Решите тригонометрические уравнения, помогите срочно, с решением через дискриминант

Answer 1

Ответ:

1) 10sin^2 x + 11sin x - 8 = 0

Замена sinx = a

10a^2 + 11a - 8 = 0

D = b^2 - 4ac = 11^2 - 4 × 10 × (-8) = 121 + 320 = 441 > 0 , 2 корня

a1 = ((-11) - √441) / 2 × 10 = ((-11)-21) / 20 = -32 / 20 = -8/5 = -1 3/5

a2 = ((-11) + 21) / 20 = 10/20 = 1/2

sin x = -1 3/5

sin x = a , a > 0

Пустое множество.

sin x = 1/2

x = (-1)^k × arcsin1/2 + πk , k € Z

x = (-1)^k × π/6 + πk , k € Z

2) 4sin^2 x - 11cosx - 11 = 0

4(1-cos^2 x) - 11cosx - 11 = 0

4 - 4cos^2 x - 11cosx - 11 = 0

-4cos^2 x - 11cosx - 7 = 0

Замена cos x = a

-4a^2 - 11a - 7 = 0

D = (-11^2) - 4 × (-4) × (-7) = 121 + 16 × (-7) = 121 - 112 = 9 > 0 , 2 корня

a1 = (11-3) / -8 = 8/-8 = -1

a2 = (11+3) / -8 = 15/-8 = -1 7/8

cos x = -1

x = π + 2πk , k € Z

cos x = -1 7/8

cos x = +-arccos(-1 7/8) + 2πk , k € Z

cos x = +- (π - arccos(1 7/8)) + 2πk , k € Z

3) 4sin^2 x + 9sin x cos x + 2cos^2 x = 0

Делим всё на cos^2 x

4tg^2 x + 9tg x + 2 = 0

Замена tg x = a

4a^2 + 9a + 2 = 0

D = 9^2 - 4 × 4 × 2 = 81 - 32 = 49

a1 = ((-9) - 7) / 8 = -16 / 8 = -2

a2 = ((-9) + 7) / 8 = -2/8 = -1/4

tg x = -2

x = arctg(-2) + πk , k € Z

x = -arctg(2) + πk , k € Z

tg x = -1/4

x = -arctg(1/4) + πk , k € Z

4) 3tg x - 8ctg x + 10 = 0

3tg x - 8/tg x + 10 = 0

Домножаем всё на tg x

3tg^2 x - 8 + 10tg x = 0

Замена tg x = a

3a^2 - 8 + 10a = 0

D = 10^2 - 4 × 3 × (-8) = 196

a1 = ((-10) - 14) / 6 = -24/6 = -4

a2 = ((-10) + 14) / 6 = 4/6 = 2/3

tg x = -4

x = -arctg(4) + πk , k € Z

tg x = 2/3

x = arctg(2/3) + πk , k € Z

5) 3sin 2x + 8sin^2 x = 7

6sinx × cosx + 8sin^2 x - 7(sin^2 x + cos^2x ) = 0

6sinxcosx + 8sin^2 x - 7sin^2 x - 7cos^2 x = 0

Делим всё на cos^2 x

6tg x + 8tg^2 x - 7tg^2 x - 7 = 0

6tg x + tg^2 x - 7 = 0

Замена tg x = a

6a + a^2 - 7 = 0

D = 6^2 - 4 × 1 × (-7) = 36 + 28 = 64

a1 = ((-6) - 8) / 2 = -14/2 = -7

a2 = ((-6) + 8) / 2 = 2/2 = 1

tg x = -7

x = -arctg(7) + πk , k € Z

tg x = 1

x = π/4 + πk , k € Z

6) 10sin^2 x + 11sin 2x + 6cos 2x = -6

10sin^2 x + 22sinxcosx + (12cos^2 x - 1) + 6(sin^2 x + cos^2 x) = 0

10sin^2 x + 22sinxcosx + (12cos^2 x - 1) + 6sin^2 x + 6cos^2 x = 0

Делим всё на cos^2 x

10tg^2 x + 22tg x + 11 + 6tg^2 x + 6 = 0

16tg^2 x + 22tg x + 17 = 0

D = отрицательный , корней нет