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!

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, $\displaystyle 2\pi$], test:

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

Given that the frequency of the cosine function is 12, you may want to do $\displaystyle \frac{\pi}{3}$, $\displaystyle \frac{\pi}{6}$, 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

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

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

The first one can be written in the form

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

which doesn't help quite so much.