Solve cos 2x+5 sinx= 2

Question

on the interval [-pie,pie]?

Answer 1

cos2x+5sinx=2

Here using formula cos2x=1-2sin^2x

1-2sin^2x+5sinx=2

2sin^2x-5sinx+1=0

To calculate roots by using formula

sinx=[-b ± sqrt(b² - 4ac)] / 2a

sinx=[5 ± sqrt(25 - 4(2)(1)] / 2(2)

     =[5 ± sqrt(25 - 8] / 4

    =[5 ± sqrt(17] / 4

Comments

sin x = [5 - sqrt(17)] / 4  and  sin x = [5 + sqrt(17)] / 4

sin x = 0.219  and  sin x = 2.281

⇒ x = sin^{- 1} (0.219)  and  x = sin^{- 1} (2.281)

x = 12.6⁰  = 12.6 * (π/180) = 0.07π.

x = sin^{- 1} (2.281) is not possible, because, the ragne of the sine function is [- 1, 1].

Therefore, there is no solution exists on the interval [- π, π].