Exact Value of sin 22½°

How to find the exact value of sin 22½° using the value of cos 45°?

Solution: 

22½° lies in the first quadrant.

Therefore, sin 22½° is positive.

For all values of the angle A we know that, cos A = 1 - 2 sin\(^{2}\) \(\frac{A}{2}\)

⇒ 1 - cos A = 2 sin\(^{2}\) \(\frac{A}{2}\)

⇒ 2 sin\(^{2}\) \(\frac{A}{2}\) = 1 - cos A

⇒ 2 sin\(^{2}\) 22½˚ = 1 - cos 45°

⇒ sin\(^{2}\) 22½˚ = \(\frac{1 - cos 45°}{2}\)

⇒ sin\(^{2}\) 22½˚ = \(\frac{1 - \frac{1}{\sqrt{2}}}{2}\), [Since we know cos 45° = \(\frac{1}{√2}\)]

⇒ sin 22½˚ = \(\sqrt{\frac{1}{2}(1 - \frac{1}{\sqrt{2}})}\), [Since, sin 22½˚ > 0]

⇒ sin 22½˚ = \(\sqrt{\frac{\sqrt{2} - 1}{2\sqrt{2}}}\)

⇒ sin 22½˚ = \(\frac{1}{2}\sqrt{2 - \sqrt{2}}\)

Therefore, sin 22½˚ = \(\frac{1}{2}\sqrt{2 - \sqrt{2}}\)

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