Exact Value of cos 7½°

How to find the exact value of cos 7½° using the value of cos 15°?

Solution: 

7½° lies in the first quadrant.

Therefore, cos 7½° is positive.

For all values of the angle A we know that, cos (α - β) = cos α cos β + sin α sin β.

Therefore, cos 15° = cos (45° - 30°) 

cos 15° = cos 45° cos 30° + sin 45° sin 30°

= \(\frac{1}{√2}\)∙\(\frac{√3}{2}\) + \(\frac{1}{√2}\)∙\(\frac{1}{2}\)

= \(\frac{√3}{2√2}\) + \(\frac{1}{2√2}\)

= \(\frac{√3 + 1}{2√2}\)

Again 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\(^{2}\) 7½˚ = 1 + cos 15°

⇒ cos\(^{2}\) 7½˚ = \(\frac{1 + cos 15°}{2}\)

⇒ cos\(^{2}\) 7½˚ = \(\frac{1 + \frac{√3 + 1}{2√2}}{2}\)

⇒ cos\(^{2}\) 7½˚ = \(\frac{2√2 + √3 + 1}{4√2}\)

⇒ cos 7½˚ = \(\sqrt{\frac{4 + √6 + √2}{8}}\), [Since cos 7½° is positive]

⇒ cos 7½˚ = \(\frac{\sqrt{4 + √6 + √2}}{2√2}\)

Therefore, cos 7½˚ = \(\frac{\sqrt{4 + √6 + √2}}{2√2}\)


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11 and 12 Grade Math

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