Find the maximum and minimum values of $A \cos t + B \sin t$

Question

Let $A$ and $B$ be constants. Find the maximum and minimum values of $A \cos t + B \sin t$.

I differentiated the function and found the solution to it as follows:

$f'(x)= B \cos t - A \sin t$

$B \cos t - A \sin t = 0 $

$t = \cot^{-1}(\frac{A}{B})+\pi n$

However, I got stuck here on how to formulate the minimum and maximum points. Any explanation would be appreciated.

Answer 1

Let $\displaystyle C = \sqrt{A^2 + B^2}$.

Then $\displaystyle A \cos t + B \sin t = C \left(\frac{A}{\sqrt{A^2+B^2}} \cos t + \frac{B}{\sqrt{A^2+B^2}} \sin t\right)$, and we can visualize a right triangle with opposite $A$ and adjacent $B$ to notice that we can find an angle $\phi$ such that

$\displaystyle \frac{A}{\sqrt{A^2+B^2}} = \sin \phi$ and $\displaystyle \frac{B}{\sqrt{A^2+B^2}} = \cos \phi$.

This means that $\displaystyle A \cos t + B \sin t = C (\sin \phi \cos t + \cos \phi \sin t) = C \sin (\phi + t)$. Now we can easily see that the maximum is $C = \sqrt{A^2+B^2}$ and the minimum is $-C = -\sqrt{A^2+B^2}$.

And of course, the maximum is reached when $\displaystyle \phi + t = \frac{\pi}{2}$, and the minimum when $\displaystyle \phi + t = \frac{3\pi}{2}$.

Answer 2

If $A=B=0$, then $$ \min_t(A\cos t+B\sin t)=\max_t(A\cos t+B\sin t)=0. $$ If $(A,B) \ne (0,0)$, let $\theta \in [0,2\pi)$ such that $$ \cos\theta=\frac{A}{\sqrt{A^2+B^2}},\quad \sin\theta=\frac{B}{\sqrt{A^2+B^2}}. $$ Then $$ A\cos t+B\sin t=\sqrt{A^2+B^2}(\cos t\cos\theta+\sin t\sin\theta)=\sqrt{A^2+B^2}\cos(t-\theta). $$ Hence $$ \min_t(A\cos t+B\sin t)=-\sqrt{A^2+B^2},\quad \max_t(A\cos t+B\sin t)=\sqrt{A^2+B^2}. $$

Answer 3

$A\cos t+ B \sin t = \sqrt{A^2+B^2} ( \frac{A}{\sqrt{A^2+B^2}} \cos t + \frac{B}{\sqrt{A^2+B^2}} \sin t)$. Choose $\theta$ such that $e^{i \theta} = \frac{A}{\sqrt{A^2+B^2}} + i\frac{B}{\sqrt{A^2+B^2}} $. Then $A\cos t+ B \sin t = \sqrt{A^2+B^2} ( \cos \theta \cos t + \sin \theta \sin t) = \sqrt{A^2+B^2} \cos(\theta-t)$.

It follows that the extreme values are $\pm \sqrt{A^2+B^2}$.