Question

Find the exact value of cos pi/12 using half angle identities

Answer 1

`\bf cos\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1+cos({{ \theta}})}{2}}\\\\ -------------------------------\\\\ \cfrac{\pi }{12}\cdot 2\implies \cfrac{\pi }{6}\qquad therefore\qquad \cfrac{\quad (\pi )/(6)\quad }{2}\implies \cfrac{\pi }{12}\qquad then \\\\\\ cos\left( (\pi )/(12) \right)\implies cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\sqrt{\cfrac{1+cos\left( (\pi )/(6) \right)}{2}}`


`\bf cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\sqrt{\cfrac{1+(√(3))/(2)}{2}}\implies cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\sqrt{\cfrac{(2+√(3))/(2)}{2}} \\\\\\ cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\sqrt{\cfrac{2+√(3)}{4}}\implies cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\cfrac{\sqrt{2+√(3)}}{√(4)} \\\\\\ cos\left( \cfrac{(\pi )/(6)}{2} \right)=\pm\cfrac{\sqrt{2+√(3)}}{2}`

Answer 2

`\cos \left((\pi )/(12)\right)=\frac{\sqrt{2+√(3)}}{2}`

Explanation:

To find the exact value of
`\cos \left((\pi )/(12)\right)` using half angle identities you must:

Write
`\cos \left((\pi )/(12)\right)` as
`\cos \left(((\pi )/(6))/(2)\right)`

Using the half angle identity
`\cos \left((x)/(2)\right)=\sqrt{(1+\cos \left(x\right))/(2)}`

`\cos \left(((\pi )/(6))/(2)\right)=\sqrt{(1+\cos \left((\pi )/(6)\right))/(2)}`

Use the following identity:
`\cos \left((\pi )/(6)\right)=(√(3))/(2)`

`\sqrt{(1+\cos \left((\pi )/(6)\right))/(2)}=\sqrt{(1+(√(3))/(2))/(2)}`

Join
`1+(√(3))/(2)`

`1+(√(3))/(2)=(1\cdot \:2)/(2)+(√(3))/(2)=(2+√(3))/(2)`

`\sqrt{(1+(√(3))/(2))/(2)}=\sqrt{((2+√(3))/(2))/(2) } =\sqrt{(2+√(3))/(4)} =\frac{\sqrt{2+√(3)}}{√(4)}=\frac{\sqrt{2+√(3)}}{2}`

Therefore,

`\cos \left((\pi )/(12)\right)=\frac{\sqrt{2+√(3)}}{2}`