solve for x. (6cos(x) +2sin(x))^2 + 32sin^2(x) = 42

Answer 1

(6cos(x)+2sin(x))² + 32sin²(x) = 42

1. 36cos²(x)+24cos(x)sin(x)+4sin²(x)+32sin²(x)=42. Multiply out the square of the sum
2. 36cos²(x)+36sin²(x) + 24cos(x)sin(x) = 42 gather like terms
3. 36+24cos(x)sin(x)=42 Use sin²(x)+cos²(x)=1
4. 24cos(x)sin(x)=6 gather like terms
5. 4cos(x)sin(x)=1 divide by 6
6. 2sin(2x)=1 use 2sin(x)cos(x)=sin(2x)
7. 2x=sin^-1(1/2) solve for argument of trig function, divide by 2
8. x=15deg (or π/6 radians) answer

Answer 2

[6cos(x) + 2sin(x) ]^2 + 32sin^2(x) = 42

36cos^2(x) + 24cos(x)sin(x) + 4sin^2(x) + 32sin^2(x) = 42

36(cos^2(x) + sin^2(x)) + 24cos(x)sin(x) = 42

36 + 24cos(x)sin(x) = 42

24cos(x)sin(x) = 6

2cos(x)sin(x) = 1/2

sin(2x) = 1/2

So...

2x = 30 degrees

x = 15 degrees