How to find the exact value of cos 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 = 2 cos\(^{2}\) \(\frac{A}{2}\) - 1
⇒ 1 + cos A = 2 cos\(^{2}\) \(\frac{A}{2}\)
⇒ 2 cos\(^{2}\) \(\frac{A}{2}\) = 1 + cos A
⇒ 2 cos\(^{2}\) 22½˚ = 1 + cos 45°
⇒ cos\(^{2}\) 22½˚ = \(\frac{1 + cos 45°}{2}\)
⇒ sin\(^{2}\) 22½˚ = \(\frac{1 + \frac{1}{\sqrt{2}}}{2}\), [Since we know cos 45° = \(\frac{1}{√2}\)]
⇒ cos 22½˚ = \(\sqrt{\frac{1}{2}(1 + \frac{1}{\sqrt{2}})}\), [Since, cos 22½˚ > 0]
⇒ cos 22½˚ = \(\sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}}\)
⇒ cos 22½˚ = \(\frac{1}{2}\sqrt{2 + \sqrt{2}}\)
Therefore, cos 22½˚ = \(\frac{1}{2}\sqrt{2 + \sqrt{2}}\)
11 and 12 Grade Math
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