Question

Marion needs to find the cosine of Pi over 12. If she knows that cos Pi over 6 = root 3 over 2, how can she use this fact to find the cosine of Pi over 12? What is her answer. Please explain and show your work

Answer 1

Use double angle formula for cosine $cos(2A)=2cos^2(A)-1$

So, the cosine of angle $A$ can be written as,

$2cos^2(A)=1+cos(2A)$

$cos^2(A)=\frac{1}{2}(1+cos(2A))$

$cos(A)=\sqrt{\frac{1}{2}(1+cos(2A))}$

Plug $A=\frac{\pi}{12}$ into the relation $cos(A)=\sqrt{\frac{1}{2}(1+cos(2A))}$ to obtain,

$cos(\frac{\pi}{12})=\sqrt{\frac{1}{2}(1+cos(2(\frac{\pi}{12})))}$

$=\sqrt{\frac{1}{2}(1+cos(\frac{\pi}{6}))}$

$=\sqrt{\frac{1}{2}(1+\frac{\sqrt{3}}{2})}$ .....plug $cos(\frac{\pi}{6})=\frac{\sqrt3}{2}$

$=\sqrt{\frac{1}{2}(\frac{2+\sqrt{3}}{2})}$

$=\sqrt{\frac{1}{4}(2+\sqrt{3})}$

$=\frac{1}{2}\sqrt{2+\sqrt{3}}\approx0.965926$

Therefore, the cosine of $\frac{\pi}{12}$ is $cos(\frac{\pi}{12})=\frac{1}{2}\sqrt{2+\sqrt{3}}\approx0.965926$