Exact Value of sin 18°

We will learn to find the exact value of sin 18 degrees using the formula of multiple angles.

How to find exact value of sin 18°?

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 = 4cos^3 A - 3 cos A

⇒ 2 sin A cos A - 4cos^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 = $$\frac{-2 \pm \sqrt{- 4 (4)(-1)}}{2(4)}$$

⇒ sin θ = $$\frac{-2 \pm \sqrt{4 + 16}}{8}$$

⇒ sin θ = $$\frac{-2 \pm 2 \sqrt{5}}{8}$$

⇒ sin θ = $$\frac{-1 \pm \sqrt{5}}{4}$$

Now sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = $$\frac{-1 \pm \sqrt{5}}{4}$$