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Math Help - Finding Exact Values

  1. #1
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    Finding Exact Values

    I have 3 problems that I need help with. All of them are 'give the exact value of the trig function'. I am totally lost here.

    cos 11pi/4

    sin 7pi/6

    tan 5pi/3
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  2. #2
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    these are all angles on the unit circle.

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  3. #3
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    \cos{\left(\frac{11\pi}{4}\right)}

    = \cos{\left(3\pi - \frac{\pi}{4}\right)}

    = \cos{(3\pi)}\,\cos{\left(\frac{\pi}{4}\right)} + \sin{(3\pi)}\,\sin{\left(\frac{\pi}{4}\right)}

    We know \sin{(n\pi)} = 0 , n\in \mathbb{I}
    And that \cos{(3\pi)} = -1

    So, \cos{(3\pi)}\,\cos{\left(\frac{\pi}{4}\right)} + \sin{(3\pi)}\,\sin{\left(\frac{\pi}{4}\right)}

    = -\cos{\left(\frac{\pi}{4}\right)} = -\frac{\sqrt{2}}{2}




    \sin{\left(\frac{7\pi}{6} \right)} = \sin{\left(\pi + \frac{\pi}{6}\right)}

    = \sin{(\pi)}\,\cos{\left(\frac{\pi}{6}\right)} + \cos{(\pi)}\,\sin{\left(\frac{\pi}{6}\right)}

    = -sin{\left(\frac{\pi}{6}\right)} = -\frac{1}{2}




    \tan{\left(\frac{5\pi}{3}\right)} = \tan{\left(2\pi - \frac{\pi}{3}\right)}

    = \frac{\tan{(2\pi)} - \tan{\left(\frac{\pi}{3}\right)}}{1 + \tan{(2\pi)}\,\tan{\left(\frac{\pi}{3}\right)}}

    =-\tan{\left(\frac{\pi}{3}\right)} = -\sqrt{3}
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  4. #4
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    Thanks nzmathman. The walkthrough of the steps is very helpful.
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