Find-the-exact-value-of-cos-12-

Question Number 133587 by bemath last updated on 23/Feb/21

Find the exact value of \cos 12 ° .

Answered by Dwaipayan Shikari last updated on 23/Feb/21

s i n \frac{π}{60} = s i n \frac{π}{10} c o s \frac{π}{12} - s i n \frac{π}{12} c o s \frac{π}{10}

= \frac{\sqrt{5} - 1}{4} . \frac{\sqrt{2 + \sqrt{3}}}{2} - \frac{\sqrt{10 - 2 \sqrt{5}}}{4} . \frac{\sqrt{2 + \sqrt{3}}}{2}

= \frac{1}{16} ((\sqrt{5} - 1) (\sqrt{6} - \sqrt{2}) - (\sqrt{6} + \sqrt{2}) \sqrt{10 - 2 \sqrt{5}})

c o s \frac{π}{60} = \sqrt{1 - {s i n}^{2} \frac{π}{60}}

c o s \frac{π}{15} = c o s \frac{π}{12} c o s \frac{π}{60} + s i n \frac{π}{60} s i n \frac{π}{12}

= \frac{\sqrt{6} + \sqrt{2}}{2} c o s \frac{π}{60} + \frac{\sqrt{6} - \sqrt{2}}{2} s i n \frac{π}{60}

= \frac{1}{8} (\sqrt{5} - 1 + \sqrt{6 (5 + \sqrt{5})})