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Exact Value of cot 7½°

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

Solution: 

7½° lies in the first quadrant.

Therefore, both sin 7½° and cos 7½° is positive.

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

Therefore, sin 15° = sin (45° - 30°)

                         = 1232 - 1212

                         = 322 - 122

                         = 3122

Again, 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°

           = 1232 + 1212

           = 322 + 122

           = 3+122

Now cot 7½°

= cos7½°sin7½°

= 2cos7½°cos7½°2sin7½°cos7½°

= 2cos27½°2sin7½°cos7½°

= 1+cos15°sin15°

= 1+cos(45°30°)sin(45°30°)

= 1+3+1223122

= 22+3+131

= (22+3+1)(3+1)(31)(3+1)

= 26+22+3+3+3+131

= 26+22+23+42

= √6 + √2 + √3 + 2

= 2 + √2 + √3 + √6

 Submultiple Angles






11 and 12 Grade Math

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