Answer:
(-1 + √5) / 4
Explanation:
18° is 1/5 of 90°. So if we say x = 18:
5x = 90
You probably won't find an identity for sin(5x) in your textbook. But you will find double angle and triple angle formulas. So if we split 5x into two terms:
2x + 3x = 90
Rearrange:
2x = 90 − 3x
Take sine of both sides:
sin(2x) = sin(90 − 3x)
Use phase shift identity:
sin(2x) = cos(3x)
Apply double and triple angle formulas:
2 sin x cos x = 4 cos³ x − 3 cos x
Simplify:
0 = 4 cos³ x − 3 cos x − 2 sin x cos x
0 = cos x (4 cos² x − 3 − 2 sin x)
We know cos 18° isn't 0, so we can divide it out:
0 = 4 cos² x − 3 − 2 sin x
Using Pythagorean identity and simplifying:
0 = 4 (1 − sin² x) − 3 − 2 sin x
0 = 4 − 4 sin² x − 3 − 2 sin x
0 = 1 − 4 sin² x − 2 sin x
0 = 4 sin² x + 2 sin x − 1
Solve with quadratic formula (or you can complete the square):
sin x = [ -2 ± √(2² − 4(4)(-1)) ] / 2(4)
sin x = [ -2 ± √(4 + 16) ] / 8
sin x = (-2 ± √20) / 8
sin x = (-2 ± 2√5) / 8
sin x = (-1 ± √5) / 4
18° is in the first quadrant, so we know sin x > 0. Therefore:
sin x = (-1 + √5) / 4