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˚ = \(\frac{1}{2}\sqrt{2 - \sqrt{2}}\)
11 and 12 Grade Math
From Exact Value of sin 22 and Half Degree to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Apr 18, 24 02:58 AM
Apr 18, 24 02:15 AM
Apr 18, 24 01:36 AM
Apr 18, 24 12:31 AM
Apr 17, 24 01:32 PM