Can someone help me out with this question: Solve: 6sin^3(x) − 13sin^2(x) + 2 sin (x) + 5 = 0

Question

Solve: 6sin^3(x) − 13sin^2(x) + 2 sin (x) + 5 = 0. Show full steps for factoring including Factor Theorem. No

synthetic division is permitted, only long division. Use exact values only.

Answer 1

let y= sinx

6y^3 -13y^2 + 2y + 5 = 0

try a simple integer solution. 0 won't work, but 1 does

y-1 is a factor

divide the cubic by y-1 using long division to get

6y^2 -7y -5 = 0. Factor to get

(3y-5)(2y+1)=0

set each factor = 0

y=5/3, y=-1/2

y=sinx = 1 and 1/2 (sines can never be greater than 1, so ignore 5/3)

x= the angles whose sines = 1 or 1/2

x = pi/2 + 2npi where n= any integer or 90 degrees + 360n or

x = pi/6 or 5pi/6 + 2npi where n = any integer or 30 degrees or 150 degrees, each + 360n degrees where n = any integer