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finlep [7]
3 years ago
14

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

Mathematics
1 answer:
skelet666 [1.2K]3 years ago
5 0

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} )

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