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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°)
= 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
(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
Now cot 7½°
= cos7½°sin7½°
= 2cos7½°∙cos7½°2sin7½°∙cos7½°
= 2cos27½°2sin7½°cos7½°
= 1+cos15°sin15°
= 1+cos(45°−30°)sin(45°−30°)
= 1+√3+12√2√3−12√2
= 2√2+√3+1√3−1
= (2√2+√3+1)(√3+1)(√3−1)(√3+1)
= 2√6+2√2+3+√3+√3+13−1
= 2√6+2√2+2√3+42
= √6 + √2 + √3 + 2
= 2 + √2 + √3 + √6
11 and 12 Grade Math
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