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Exact Value of sin 7½°

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

Solution: 

7½° lies in the first quadrant.

Therefore, sin 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 = 1 - 2 sin2A2

⇒ 1 - cos A = 2 sin2 A2

⇒ 2 sin2 A2 = 1 - cos A

⇒ 2 sin2 7½˚ = 1 - cos 15°

⇒ sin2 7½˚ = 1−cos15°2

⇒ sin2 7½˚ = 1−√3+12√22

⇒ sin2 7½˚ = 2√2−√3−14√2

⇒ sin 7½˚ = √4−√6−√28, [Since sin 7½° is positive]

⇒ sin 7½˚ = √4−√6−√22√2

Therefore, sin 7½˚ = √4−√6−√22√2



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11 and 12 Grade Math

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