Ranges in trigonometry

Question

How to find the range of the sum or difference of two trigonometric functions?

$2\sin x-3\cos x$

Before this whenever the question of range i have solved they were either single trigonometric function or if they were in pair then they were in form of squires (eg. $2\cos^2x + \sin^2x$) so i used to convert them into one function.

I don't know how to solve this type of problem, i think we here also we need to change the it into single function?

That would depend on what trigonometric ratios you wanted to find the range of. — JimmyK4542, Aug 16 '14 at 02:26
If it is $\sin\theta+\cos\theta$, either use the fact that this is $\sqrt{2}\sin(\theta+\pi/4)$, or work with $(\sin\theta+\cos\theta)^2$. — André Nicolas, Aug 16 '14 at 02:44
Also, assuming these functions are continuous functions $\mathbb R\rightarrow \mathbb R$, the "range" (or what I would more likely call the "image") will be an interval since it is the continuous image of a connected space and lies in $\mathbb R$. — MPW, Aug 16 '14 at 03:21
Can you be more specific? What particular sum did you have in mind? — MPW, Aug 16 '14 at 03:24
Yes you should always provide all the necessary information. I hope soon this question will be opened! — Freddy, Aug 16 '14 at 04:29
While we are waiting on the 5th reopen vote, I suggest reading this. — JimmyK4542, Aug 16 '14 at 05:16

score 5 · Accepted Answer · edited Jun 12 '20 at 10:38

Let $\theta$ be an angle which satisfies $\cos \theta = \dfrac{2}{\sqrt{13}}$ and $\sin \theta = -\dfrac{3}{\sqrt{13}}$. Then we have:

$2\sin x - 3\cos x$

$= \sqrt{13}\left(\dfrac{2}{\sqrt{13}}\sin x - \dfrac{3}{\sqrt{13}}\cos x\right)$

$= \sqrt{13}\left(cos\theta\sin x + \sin\theta\cos x\right)$

$= \sqrt{13}\sin(x+\theta)$.

Do you know what the range of $\sqrt{13}\sin(x+\theta)$ is?

Using the same method, we can get the general result $A\sin x + B\cos x = \sqrt{A^2+B^2}\sin(x+\theta)$ where $\theta$ satisfies $\cos\theta = \dfrac{A}{\sqrt{A^2+B^2}}$ and $\sin\theta = \dfrac{B}{\sqrt{A^2+B^2}}$. See this pdf for details.

Thanks Freddy for suggesting to include that link in my answer.

Ranges in trigonometry

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