Answer:
sin 18° = sin A = −1±5√4
Explanation:
Let A = 18°
- Therefore, 5A = 90°
2A + 3A = 90˚
2θ = 90˚ - 3A
- Taking sine on both sides, we get
sin 2A = sin (90˚ - 3A) = cos 3A
2 sin A cos A = 4 cos^3 A - 3 cos A
2 sin A cos A - 4 cos^3A + 3 cos A = 0
cos A (2 sin A - 4 cos^2 A + 3) = 0
- Dividing both sides by cos A = cos 18˚ ≠ 0, we get
2 sin θ - 4 (1 - sin^2 A) + 3 = 0
4 sin^2 A + 2 sin A - 1 = 0, which is a quadratic in sin A
- Therefore, sin θ = −2±−4(4)(−1)√2(4)
sin θ = −2±4+16√8
sin θ = −2±25√8
sin θ = −1±5√4
- Now sin 18° is positive, as 18° lies in first quadrant.
- Therefore, sin 18° = sin A = −1±5√4
(Good luck! I hope this helped. ^_^)