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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°

= 1√2∙√32 + 1√2∙12

= √32√2 + 12√2

= √3+12√2

Again for all values of the angle A we know that, cos A = 2 cos2A2  - 1

⇒ 1 + cos A = 2 cos2 A2

⇒ 2 cos2 A2 = 1 + cos A

⇒ 2 cos2 7½˚ = 1 + cos 15°

⇒ cos2 7½˚ = 1+cos15°2

⇒ cos2 7½˚ = 1+√3+12√22

⇒ cos2 7½˚ = 2√2+√3+14√2

⇒ cos 7½˚ = √4+√6+√28, [Since cos 7½° is positive]

⇒ cos 7½˚ = √4+√6+√22√2

Therefore, cos 7½˚ = √4+√6+√22√2


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11 and 12 Grade Math

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