1. ## Equivalent Trig Expressions

Given that tan 4pi/9=5.6713, determine the following, to four decimal places, without using a calculator. a)cot(pi/18) b)tan(13pi/9)

I got a by using a cofunction identity:
cot(pi/18)= tan(pi/2-pi/18)
=tan(4pi/9)
=5.6713

But I don't know how to do b. What formula do I use?

2. ## Re: Equivalent Trig Expressions

Hello, Dragon08!

Given that $\displaystyle \tan \tfrac{4\pi}{9}\,=\,5.6713$,
determine the following to four decimal places, without using a calculator.

. . $\displaystyle (a)\;\cot\left(\tfrac{\pi}{18}\right)$
We need this identity: .$\displaystyle \cot(A - B) \:=\:\frac{1 + \cot A\cot B}{\cot B - \cot A}$

$\displaystyle \cot\left(\tfrac{\pi}{18}\right) \;=\;\cot\left(\tfrac{\pi}{2} - \tfrac{4\pi}{9}\right) \;=\;\frac{1 + \cot\left(\frac{\pi}{2}\right)\cot\left(\frac{4\pi }{9}\right)} {\cot\left(\frac{4\pi}{9}\right) - \cot\left(\frac{\pi}{2}\right)}$

. . . . . . $\displaystyle =\;\frac{1 + 0\cdot\cot\left(\frac{4\pi}{9}\right)}{\cot\left({ 4\pi}\over{9}\right) - 0} \;=\;\frac{1}{\cot\left(\frac{4\pi}{9}\right)} \;=\;\tan\left(\tfrac{4\pi}{9}\right) \;=\;5.6713$

$\displaystyle (b)\;\tan\left(\tfrac{13\pi}{9}\right)$

We need this identity: .$\displaystyle \tan(A + B) \;=\;\frac{\tan A + \tan B}{1 - \tanA\tan B}$

$\displaystyle \tan\left(\tfrac{13\pi}{9}\right) \;=\;\tan\left(\pi + \tfrac{4\pi}{9}\right) \;=\; \frac{\tan(\pi) + \tan\left(\frac{4\pi}{9}\right)}{1 - \tan(\pi)\tan\left(\frac{4\pi}{9}\right)}$

. . . . . . $\displaystyle =\;\frac{0 + \tan\left(\frac{4\pi}{9}\right)}{1 - 0\cdot\tan\left(\frac{4\pi}{9}\right)} \;=\; \tan\left(\tfrac{4\pi}{9}\right) \;=\;5.6713$

Thank you!