Question

The value of sin 18° is ______.

विकल्प

• `(sqrt5 + 1)/4`

• `(sqrt5 - 1)/4`

• `4/(sqrt5 + 1)`

• `4/(sqrt5 - 1)`

Answer 1

The value of sin 18° is `underline((sqrt5 - 1)/4)`.

Explanation:

Let, θ = 18°

then 2θ = 36°= 90°- 54° = 90°- 3θ

Now, sin2θ = sin(90°- 3θ)

⇒ 2 sin θ cos θ = cos 3θ

⇒ 2 sin θ cos θ = 4 cos³ θ - 3 cos θ

⇒ 2 sin θ cos θ = cos θ (4 cos² θ - 3)

⇒ 2 sin θ = 4 - 4 sin² θ - 3

⇒ 4 sin² + θ + 2 sin θ - 1 = 0

⇒ sin θ = `(- 2 +- sqrt(4 + 16))/(2(4)) = (- 1 +- sqrt5)/4`

⇒ But as sin θ > 0 we have sin θ > 0 we have

sin θ = `(sqrt5 - 1)/4`

∴ sin 18° = `(sqrt5 - 1)/4`