Question

40 POINTS!

Find the exact value by using a half-angle identity.

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Answer 1

I agree with him because it's correct and makes sense!

Answer 2

`\cos(5\pi)/(12)=\cos\left((6\pi)/(12)-(\pi)/(12)\right)=\cos\left((\pi)/(2)-(\pi)/(12)\right)=(*)`


`Use: \cos(\alpha-\beta)=\cos\alpha \cos\beta+\sin\alpha \cos\beta`


`(*)=\cos(\pi)/(2) \cos(\pi)/(12)+\sin(\pi)/(2) \sin(\pi)/(12)\\\\=0\cdot\cos(\pi)/(12)+1\cdot\sin(\pi)/(12)=\sin(\pi)/(12)`


`(\pi)/(12)=((\pi)/(6))/(2)`


`The\ half-angle\ identity\ \sin^2(\alpha)/(2)=(1)/(2)\left(1-\cos\alpha\right)`


`Using\ the\ above\ formulas,\ we\ get:\\\\\sin^2(\pi)/(12)=(1)/(2)\left(1-\cos(\pi)/(6)\right)\\\\\sin^2(\pi)/(12)=(1)/(2)\left(1-(\sqrt3)/(2)\right)\\\\\sin^2(\pi)/(12)=(1)/(2)-(\sqrt3)/(4)`


`Since\ 0 \ \textless \ (\pi)/(12) \ \textless \ \pi,\ then\ \sin(\pi)/(12)\ is\ a\ positive\ number.\\\\ Therefore,\ we\ have:\\\\\sin(\pi)/(12)=\sqrt{(1)/(2)-(\sqrt3)/(4)}=\sqrt{(2-\sqrt3)/(4)}=(√(2-\sqrt3))/(2)`


`Answer:\ \boxed{\cos(5\pi)/(12)=(√(2-\sqrt3))/(2)}`