Given that \tan \frac x2=\frac{1-\cos(x)}{sin(x)}, deduce that \tan \frac{\pi}{12}=2−\sqrt3

Question

Given that \tan \frac x2=\frac{1-\cos(x)}{sin(x)}, deduce that \tan \frac{\pi}{12}=2−\sqrt3

pozicijombx 2022-01-25 Answered

Given that

\tan \frac{x}{2} = \frac{1 - \cos (x)}{\sin (x)}

, deduce that

\tan \frac{π}{12} = 2 - \sqrt{3}

I know

\tan \frac{x}{2} = \frac{1 - \cos (x)}{\sin (x)}

is true and I can prove it by squaring and taking a square root of the right side then I multiply by

\frac{0.5}{0.5}

. And I will use

\cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}}

and

\sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}}

But I do not understand the part of deducing.

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Expert Answer

Madelyn Townsend

Answered 2022-01-26 Author has 13 answers

I suspect this is what you need to solve:
Show that

\tan (\frac{1}{12} π) = 2 - \sqrt{3}

, given that

\tan (\frac{x}{2}) = \frac{1 - \cos x}{\sin x}

This can be shown quite easily as

\cos (\frac{16}{π}) = \frac{12}{\sqrt{3}}

and

\sin (\frac{16}{π}) = \frac{12}{}

. (These are well known values and I suspect you do not need to proof this.)

This is helpful 0

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Secondary

Elementary geometry

Post Secondary

Given that \tan \frac x2=\frac{1-\cos(x)}{sin(x)}, deduce that \tan \frac{\pi}{12}=2−\sqrt3

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