Question

Use a half -angle formula to find sin pi/8 exactly.

Answer 1

`\sin \left((\pi )/(8)\right)=\frac{\sqrt{2-√(2)}}{2}`

Explanation:

To find the exactly value of
`\sin \left((\pi )/(8)\right)` you must:

Step 1: Write
`\sin \left((\pi )/(8)\right)` as
`\sin \left(((\pi )/(4))/(2)\right)`

Step 2: Use the half angle identity
`\sin \left((x)/(2)\right)=\sqrt{(1-\cos \left(x\right))/(2)}`

`\sqrt{(1-\cos \left((\pi )/(4)\right))/(2)}`

Step 3: Use the following identity
`\cos \left((\pi )/(4)\right)=(√(2))/(2)`

`\sqrt{(1-(√(2))/(2))/(2)}`

Step 4: Simplify

`(1-(√(2))/(2))/(2)=(2-√(2))/(4)\\\\\sqrt{(2-√(2))/(4)}\\\\\mathrm{Apply\:radical\:rule\:}\sqrt[n]{(a)/(b)}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\\\\\frac{\sqrt{2-√(2)}}{√(4)}\\\\\frac{\sqrt{2-√(2)}}{2}`