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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.

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    $\begingroup$ If the max of $\sin3\theta$ is $1$, and the max of $\cos2\theta$ is $1$, then what is the maximum of their products? $\endgroup$ – Andrew Chin Nov 14 '19 at 19:18
  • $\begingroup$ 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. $\endgroup$ – Octavio Berlanga Nov 14 '19 at 19:28
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    $\begingroup$ @AndrewChin This observation does not work when maximizing $\sin(2\theta)\cos(3\theta)$, for example. A more rigorous argument is needed. $\endgroup$ – WE Tutorial School Nov 14 '19 at 19:45
  • $\begingroup$ 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 $\pm1$. $\endgroup$ – WE Tutorial School Nov 14 '19 at 19:54
  • $\begingroup$ @WETutorialSchool The other possibility is that $\alpha/\beta$ is an irrational number. $\endgroup$ – user Nov 14 '19 at 21:47
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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 $\cos\theta=0$.

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  • $\begingroup$ Thanks a lot, that was very useful. @Quanto $\endgroup$ – Octavio Berlanga Dec 27 '19 at 17:52
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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 $$|\sin y\cos z|\le 1.$$

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