How do you find the exact values of sin 22.5° using the half-angle formula?

We can find the exact value by using the half-angle formula.

Answer: The exact value of sin 22.5° using the half-angle formula is √(2 - √2) / 2 or 0.38

Let us proceed step by step to find the exact values of sin 22.5° using the half-angle formula.

Explanation:

Let us consider θ = 22.5 º

Therefore, sin²θ =  [(1− cos 2θ) / 2] -------(1)

If θ = 22.5° then 2θ is 45°.

On substituting the value of θ in equation (1), we get

sin² 22.5° =  [(1− cos (2 × 22.5°)) / 2]

sin² 22.5° =  [(1− cos 45°) / 2]

sin² 22.5°=  [ [1 - (1 / √2)] / 2 ]

sin² 22.5°=  (2 - √2) / 4 [ on simplifying the above RHS value ]

sin 22.5° = [√(2 - √2) / 2] 

After substituting the value of √2 as 1.414 in the above equation, we get

sin 22.5°= 0.38 

Hence, the exact value of sin 22.5° using the half-angle formula is  √(2 - √2) / 2 or  0.38.