solve: 3sin^2x-8sinx-3=0, where 0<x<2pi. the answer key said the answers are x=3.4814 and 5.9433. I understand how to get the 3.48, but how do you get the 5.94?

Answer 1

let y = sinx

3y^2 - 8y - 3 = 0

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

y = -1/3 or 3 = sinx

x = the angle whose sine is -1/3 or 3

but -1 < sinx < 1 so ignore 3

sinx = -1/3 with x in the 3rd or 4th quadrant where sines < 0

use an inverse trig function on a calculator

x = about 19.47 for the reference angle

add 180 to get x = 199.47 degrees = 199.47(pi/180) = 3.481 radians

and

subtract from 360 to get 340.53 degrees = 340.53(pi/180) = 5.943 radians

Answer 2

The angle you found, 3.4814 is in the 3rd quadrant. Sin is also negative in the 4th quadrant so there should also be a 4th quadrant angle that will be a solution between 0 and 2π. If you find the reference angle, 3.481-3.1416 (π), you get 0.3394. The 4th quadrant angle would be 2π-0.3394 = 5.9434