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Math Help - Equivalent Trig Expressions

  1. #1
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    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?
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  2. #2
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    Re: Equivalent Trig Expressions

    Hello, Dragon08!

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

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

    \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)}

    . . . . . . =\;\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




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

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

    \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)}

    . . . . . . =\;\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

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  3. #3
    Junior Member
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    Re: Equivalent Trig Expressions

    Thank you!
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