Question

The value of `sin  π/10` is ______.

विकल्प

• `(-1 + sqrt(5))/4`

• `(2 - sqrt(3))/2`

• `(2 + sqrt(3))/2`

• `(1 - sqrt(5))/4`

Answer 1

The value of `sin  π/10` is `underlinebb((-1 + sqrt(5))/4)`.

Explanation:

`π/10` = 18°

Let, θ = 18°, 2θ = 36°, 3θ = 54°

We have, 2θ + 3θ = 90°

2θ = 90° – 3θ

∴ sin 2θ = sin(90° – 3θ)

2 sin θ . cos θ = cos 3θ

∴ 2 sin θ . cos θ = 4 . cos³ θ – 3 cos θ

2 sin θ = 4 . cos² θ – 3

2 sin θ = 4(1 – sin² θ) – 3

2 sin θ = 4 – 4 sin² θ – 3

4 sin² θ + 2 sin θ – 1 = 0

∴ sin θ = `(- 2 ± sqrt(4 + (4)(4)(1)))/(2(4)) = (-2 ± 2sqrt(5))/(2(4))`

∴ sin θ = `(-1 ± sqrt(5))/4`

∴ sin θ = `(-1 + sqrt(5))/4`  ...[∵ θ is an acute angle]

∴ sin 18° = `(-1 + sqrt(5))/4`