1) 2sin^2x + 3sinx = 2 2) tg^2x - 4tgx + 3 = 0 3) 2 cos^2x - 5cosx = 3 4) 2sin^2x+sunx=0 5) 6sin^2x - 2sin2x=1 6) sin^2x - cos^2x = 7) 4sinx*cosx = 8) cos^4x - sin^4x=1

Question

София Борисова

Математика

24 июля, 11:10

1) 2sin^2x + 3sinx = 2 2) tg^2x - 4tgx + 3 = 0 3) 2 cos^2x - 5cosx = 3 4) 2sin^2x+sunx=0 5) 6sin^2x - 2sin2x=1 6) sin^2x - cos^2x = 7) 4sinx*cosx = 8) cos^4x - sin^4x=1

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Сванчо · Answer 1

1) 2sin²x + 3sinx - 2 = 0.

sinx = a.

2a² + 3a - 2 = 0.

D = 3² - 4 • 2 • (-2) = 9 + 16 = 25 = 5².

a' = - 3 + 5 / 4 = ½.

a" = - 3 - 5 / 4 = - 2.

sinx' = ( - 1) ^k • π / 6 + πk. sinx" = ( - 1) ^k + 1 • arcsin2 + πk.

2) tg²x - 4tgx + 3 = 0.

tgx = z.

z² - 4z + 3=0.

D = - 4² - 4 • 1 • 3 = 16 - 12 = 4 = 2².

z' = 4 + 2 / 2 = 3. z" = 4 - 2 / 2 = 1.

tgx' = arctg3 + πk. tgx" = π / 4 + πk.

3) 2cos²x - 5cosx - 3 = 0.

D = - 5² - 4 • 2 • ( - 3) = 25 + 24 = 49 = 7².

cosx' = 5 + 7 / 4 = 3. cosx" = 5 - 7 / 4 = - ½.

x' = ± arccos3 + 2πk. x" = 2π / 3 + 2πk.

4) 2sin²x + sinx = 0.

sinx (2sinx + 1) = 0.

sinx = 0. 2sinx + 1 = 0.

x' = πk. sinx = - ½.

x" = 5π / 6 + πk.

5) 6sin²x - 2sin2x = 1.

6sin²x - 4sinxcosx - cos²x - sin²x = 0.

5sin²x - 4sinxcosx - cos²x = 0 (: cos²x).

5tg² - 4tg - 1 = 0.

D = - 4² - 4 • 5 • ( - 1) = 16 + 20 = 36 = 6².

tgx = 4 + 6 / 10 = 1. tgx = 4 - 6 / 10 = - 0,2.

x' = π / 4 + πk. x" = - arctg0,2 + πk.

6) sin²x - cos²x = 0.

sin²x - 1 + sin²x = 0.

2sin²x = 1.

sin²x = ½.

sinx = ± √2 / 2.

x' = ( - 1) ^k • π / 4 + πk. x" = ( - 1) ^k • 3π / 4 + πk.

7) 4sinxcosx = 0.

2sin2x = 0.

sin2x = 0.

2x = πk.

x = πk / 2.

8) cos⁴x - sin⁴x = 1.

cos4x = 1.

4x = 2πk.

x = πk / 2.