# Find maximum and minimum of $\sin(3\theta)\cos(2\theta)$

find the maximum and minimum values of $$r=\sin(3\theta)\cos(2\theta)$$

I’m studying calculus, the chapter in my book is about graphing in polar coordinates.

In order to find the maximum and minimum values of $$r$$ I need to get a value of $$\theta$$ for which $$r’=-2\sin(3\theta)\sin(2\theta) + 3\cos(3\theta)\cos(2\theta)=0$$ According to my book the answer is max $$=1$$ And min $$=-1$$ I’m not sure how to get this, could anyone help me?

Note. I already tried with some trigonometric identities for reducing the equation but cannot get anywhere.

• If the max of $\sin3\theta$ is $1$, and the max of $\cos2\theta$ is $1$, then what is the maximum of their products? – Andrew Chin Nov 14 '19 at 19:18
• That’s a very good point, I just thought I could get the same result with the derivative of $r$, don’t know why it failed. – Octavio Berlanga Nov 14 '19 at 19:28
• @AndrewChin This observation does not work when maximizing $\sin(2\theta)\cos(3\theta)$, for example. A more rigorous argument is needed. – WE Tutorial School Nov 14 '19 at 19:45
• This problem is special. If you want to optimize $f:\mathbb{R}\to\mathbb{R}$ where $$f(\theta)= \sin(\alpha\theta)\cos(\beta\theta)$$ and $\alpha,\beta\in\mathbb{R}$ are such that $\frac{\alpha}{\beta}$ is a rational number of the form $\frac{p}{q}$ where $p$ is odd and $q$ is even, then the maximum and the minimum of $f$ are $1$ and $-1$ respectively. This is the only case where Andrew Chin's hint is useful. For other pairs $(\alpha,\beta)$, the optimum values are not $\pm1$. – WE Tutorial School Nov 14 '19 at 19:54
• @WETutorialSchool The other possibility is that $\alpha/\beta$ is an irrational number. – user Nov 14 '19 at 21:47

You may use the double and triple angle identifies for both sine and cosine functions to factorize your derivative equation below

$$-2\sin(3\theta)\sin(2\theta) + 3\cos(3\theta)\cos(2\theta)=0$$

into the form

$$\cos\theta(10\cos^22\theta-5\cos2\theta-2)=0$$

The solution to the factor $$\cos\theta=0$$ is just $$\theta=\frac\pi2+n\pi$$. The second factor is quadratic in $$\cos2\theta$$ and also be readily solved.

Then, you may compare the extrema values at those points and conclude that both the maximum (+1) and the minimum (-1) come from the factor $$\cos\theta=0$$.

• Thanks a lot, that was very useful. @Quanto – Octavio Berlanga Dec 27 '19 at 17:52

Hint. Simply note that $$|\sin y|\le 1$$ and $$|\cos z|\le 1$$ for all real values of $$y,z.$$ Then multiply both sides of the first by LHS of second to obtain $$|\cos z||\sin y|\le |\cos z|\le 1,$$ or that $$|\sin y\cos z|\le 1.$$