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Slav-nsk [51]
3 years ago
8

Find \tan\left(\frac{17\pi}{12}\right)tan( 12 17π ​ )tangent, left parenthesis, start fraction, 17, pi, divided by, 12, end frac

tion, right parenthesis exactly using an angle addition or subtraction formula.
Mathematics
1 answer:
uranmaximum [27]3 years ago
7 0

One way to do this is to notice

\dfrac{17\pi}{12}=\dfrac\pi6+\dfrac{5\pi}4

Then

\tan\dfrac{17\pi}{12}=\tan\left(\dfrac\pi6+\dfrac{5\pi}4\right)=\dfrac{\tan\frac\pi6+\tan\frac{5\pi}4}{1-\tan\frac\pi6\tan\frac{5\pi}4}

We have

\tan\dfrac\pi6=\dfrac{\sin\frac\pi6}{\cos\frac\pi6}=\dfrac{\frac12}{\frac{\sqrt3}2}=\dfrac1{\sqrt3}

and since \tan x has a period of \pi,

\tan\dfrac{5\pi}4=\tan\left(\pi+\dfrac\pi4\right)=\tan\dfrac\pi4=1

and so

\tan\dfrac{17\pi}{12}=\dfrac{\frac1{\sqrt3}+1}{1-\frac1{\sqrt3}}=\dfrac{1+\sqrt3}{\sqrt3-1}=2+\sqrt3

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