Anish Buchanan

2021-08-11

Use the sum and difference identites to find the exact values of sine, cosine, tangent of the angle $\frac{7\pi }{12}$

Bella

Use of sim and difference identity:
$\mathrm{cos}\left(\frac{7\pi }{12}\right)=\mathrm{cos}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)=\mathrm{cos}\frac{\pi }{4}\mathrm{cos}\frac{\pi }{3}-\mathrm{sin}\frac{\pi }{4}\cdot \mathrm{sin}\frac{\pi }{3}$
$=\frac{\sqrt{2}}{2}×\frac{1}{2}-\frac{\sqrt{2}}{2}×\frac{\sqrt{3}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}=-0.2588$
$\mathrm{sin}\left(\frac{7\pi }{12}\right)=\mathrm{sin}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)=\mathrm{sin}\frac{\pi }{4}\cdot \mathrm{cos}\frac{\pi }{3}+\mathrm{cos}\frac{\pi }{4}\mathrm{sin}\frac{\pi }{3}$
$=\frac{\sqrt{2}}{2}×\frac{1}{2}+\frac{\sqrt{2}}{2}×\frac{\sqrt{3}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}=0.9659$
$\mathrm{tan}\left(\frac{7\pi }{12}\right)=\mathrm{tan}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)$
$=\frac{\mathrm{tan}\frac{\pi }{4}+\mathrm{tan}\frac{\pi }{3}}{1-\mathrm{tan}\frac{\pi }{4}\cdot \mathrm{tan}\frac{\pi }{3}}=\frac{1+\sqrt{3}}{1-\sqrt{3}}=-3.732$

Jeffrey Jordon