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Math Help - how to sketch a graph of the product of two trig identities?

  1. #1
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    how to sketch a graph of the product of two trig identities?

    Hey all,

    so, I have encountered a problem that asks me to sketch the graph of the following: f(x) = sin(x)cos(12x). I know how to sketch them both separately, but what is the procedure so that I can do this properly without using a calculator?

    another type of problem asks me to sketch something like this: 4cos(x)-3sin(x). I have the same issue here as I would the above example.


    thanks in advance for your help!
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  2. #2
    Junior Member Bradyns's Avatar
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    Re: how to sketch a graph of the product of two trig identities?

    Test the f(x) values incrementally, to get a rough idea on the structure of the graph.

    If this is between [0, 2\pi], test:

    0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi.

    Given that the frequency of the cosine function is 12, you may want to do \frac{\pi}{3}, \frac{\pi}{6}, etc as well..

    Essentially, do what you did in high school when you first learnt how to graph linear equations, have a row of values for x, and a row for f(x).

    Of course, there is always Wolfram|Alpha; it can give you some guidance.
    sin(x)cos(12x) - Wolfram|Alpha
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  3. #3
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    Re: how to sketch a graph of the product of two trig identities?

    Additionally, you might make use of trig identities, when convenient.

    Your second example, for example, can be written in the form

    5\left(\frac{4}{5}\cos x - \frac{3}{5}\sin x \right)=5\cos (x +\alpha), \text{  where  } \alpha = \tan^{\, -1}(3/4).

    The first one can be written in the form

    \frac{1}{2}\left( \sin \frac{13}{2}x - \sin \frac{11}{2}x \right)

    which doesn't help quite so much.
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