Question

What is the exact value of Cosine StartFraction 7 pi Over 12 EndFraction?

Answer 1

Answer:

$\frac{1}{4}$ ($\sqrt{2}$ - $\sqrt{6}$ )

Step-by-step explanation:

Using the addition formula for cosine

cos(a + b) = cosacosb - sinasinb

and the exact values

cos$\frac{\pi }{3}$ = $\frac{1}{2}$, sin$\frac{\pi }{3}$ = $\frac{\sqrt{3} }{2}$ , cos$\frac{\pi }{4}$ = sin$\frac{\pi }{4}$ = $\frac{\sqrt{2} }{2}$

Note $\frac{7\pi }{12}$ = $\frac{\pi }{3}$ + $\frac{\pi }{4}$ , thus

cos$\frac{7\pi }{12}$ = cos($\frac{\pi }{3}$ + $\frac{\pi }{4}$ )

= cos$\frac{\pi }{3}$cos$\frac{\pi }{4}$ - sin$\frac{\pi }{3}$sin$\frac{\pi }{4}$

= ( $\frac{1}{2}$ × $\frac{\sqrt{2} }{2}$ ) - ($\frac{\sqrt{3} }{2}$ × $\frac{\sqrt{2} }{2}$ )

= $\frac{\sqrt{2} }{4}$ - $\frac{\sqrt{6} }{4}$

= $\frac{1}{4}$ ($\sqrt{2}$ - $\sqrt{6}$ )