Explanation:
cos(3x) = sin(2x)
Use reflection.
sin(π/2 − 3x) = sin(2x)
π/2 − 3x = 2x
π/2 = 5x
x = π/10
Instead, use multiple angle formulas.
cos(3x) = sin(2x)
4 cos³x − 3 cos x = 2 sin x cos x
4 cos³x − 3 cos x − 2 sin x cos x = 0
cos x (4 cos²x − 3 − 2 sin x) = 0
cos x (4 (1 − sin²x) − 3 − 2 sin x) = 0
cos x (4 − 4 sin²x − 3 − 2 sin x) = 0
-cos x (-4 + 4 sin²x + 3 + 2 sin x) = 0
-cos x (4 sin²x + 2 sin x − 1) = 0
If t = sin(π/10), then:
4t² + 2t − 1 = 0
Solve with quadratic formula:
t = [ -2 ± √(2² − 4(4)(-1)) ] / 2(4)
t = [ -2 ± √(4 + 16) ] / 8
t = (-2 ± 2√5) / 8
t = (-1 ± √5) / 4
Since π/10 is in the first quadrant, sin(π/10) is positive.
sin(π/10) = (-1 + √5) / 4