Answer:
[tex]\sin \left(\frac{\pi }{8}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}[/tex]
Step-by-step explanation:
To find the exactly value of [tex]\sin \left(\frac{\pi }{8}\right)[/tex] you must:
Step 1: Write [tex]\sin \left(\frac{\pi }{8}\right)[/tex] as [tex]\sin \left(\frac{\frac{\pi }{4}}{2}\right)[/tex]
Step 2: Use the half angle identity [tex]\sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos \left(x\right)}{2}}[/tex]
[tex]\sqrt{\frac{1-\cos \left(\frac{\pi }{4}\right)}{2}}[/tex]
Step 3: Use the following identity [tex]\cos \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}[/tex]
[tex]\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}[/tex]
Step 4: Simplify
[tex]\frac{1-\frac{\sqrt{2}}{2}}{2}=\frac{2-\sqrt{2}}{4}\\\\\sqrt{\frac{2-\sqrt{2}}{4}}\\\\\mathrm{Apply\:radical\:rule\:}\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\\\\\frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}\\\\\frac{\sqrt{2-\sqrt{2}}}{2}[/tex]
