# How do you find the exact value of cos(pi/4+pi/3)

Tia English 2022-09-30 Answered
How do you find the exact value of $\mathrm{cos}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)$?
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Haylie Campbell
In order to evaluate $\mathrm{cos}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)$ we need to use the compound angle formula for $\mathrm{cos}:\mathrm{cos}\left(A+B\right)\equiv \mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$
$\therefore \mathrm{cos}\left(\frac{\pi }{4}+\frac{\pi }{3}\right)=\mathrm{cos}\left(\frac{\pi }{4}\right)\mathrm{cos}\left(\frac{\pi }{3}\right)-\mathrm{sin}\left(\frac{\pi }{4}\right)\mathrm{sin}\left(\frac{\pi }{3}\right)=\frac{1}{\sqrt{2}}\ast \frac{1}{2}-\frac{1}{\sqrt{2}}$
$\ast \frac{\sqrt{3}}{2}=\frac{1}{2\sqrt{2}}-\frac{\sqrt{3}}{2\sqrt{2}}=\frac{1-\sqrt{3}}{2\sqrt{2}}=\frac{\sqrt{2}-\sqrt{6}}{4}$