# Exact Value of sin 54°

We will learn to find the exact value of sin 54 degrees using the formula of submultiple angles.

How to find the exact value of sin 54°?

Solution:

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$$^{3}$$ A + 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 = $$\frac{-1 \pm \sqrt{5}}{4}$$

Now, cos 36° = cos 2 ∙ 18°

⇒ cos 36° = 1 - 2 sin$$^{2}$$ 18°

⇒ cos 36° = 1 - 2$$(\frac{\sqrt{5} - 1}{4})^{2}$$

⇒ cos 36° = $$\frac{16 - 2(5 + 1 - 2\sqrt{5})}{16}$$

⇒ cos 36° = $$\frac{1 + 4\sqrt{5}}{16}$$

⇒ cos 36° = $$\frac{\sqrt{5} + 1}{4}$$

Now sin 54° = sin (90° - 36°) = cos 36° = $$\frac{√5 + 1}{4}$$

Therefore, sin 54° = $$\frac{√5 + 1}{4}$$

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