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Math Help - Trigonometric and Hyperbolic identities

  1. #1
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    Trigonometric and Hyperbolic identities

    Question is as follows:-

    Hence solve for t for values in the range 0 ≤ t ≤ 2 π rad:
    5.5 Cos t + 7.8 Sin t = 4.5

    Any help would be greatful, thanks
    Last edited by boza100; November 23rd 2012 at 02:25 PM.
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Trigonometric and Hyperbolic identities

    Quote Originally Posted by boza100 View Post
    Question is as follows:-

    Hence solve for t for values in the range 0 ≤ t ≤ 2 π rad:
    5.5 Cos t + 7.8 Sin t = 4.5

    Any help would be greatful, thanks
    Well it's a bit ugly but one approach would be to recall that cos(t) = \sqrt{1 - sin^2(t)} and plug that in...
    5.5~\sqrt{1 - sin^2(t)} + 7.8~sin(t) = 4.5

    Isolate the radical and square it. This will give you a quadratic in sin(t) which you can solve using the quadratic formula. Check for extra, but not valid, solutions.

    Also since in reality cos(t) = \pm \sqrt{1 - sin^2(t)} you should also work through the negative solution as well. And again check your solutions with the original equation.

    -Dan
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    MHF Contributor MarkFL's Avatar
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    Re: Trigonometric and Hyperbolic identities

    Another approach would be to use a linear combination identity to write the equation as:

    \sqrt{5.5^2+7.8^2}\sin\left(t+\tan^{-1}\left(\frac{5.5}{7.8} \right) \right)=4.5

    \sin\left(t+\tan^{-1}\left(\frac{55}{78} \right) \right)=\frac{45}{\sqrt{9109}}

    Now, after finding the quadrant IV solution, use the identity \sin(\pi-x)=\sin(x) to get the quadrant III solution.
    Last edited by MarkFL; November 23rd 2012 at 03:21 PM.
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