Just Observed A Cool Result!

I just found a generalised formula to find the range of the functions of the form: \[f(x)=a\cos x + b\sin x + c\]

The range of the above function would be: $\left[-\sqrt{a^2+b^2}+c, \sqrt{a^2+b^2}+c\right]$ .

Here's what I did;

1. Multiply and divide the function by $\sqrt{a^2 + b^2}$ ;

$f(x)=\sqrt{a^2 + b^2}\left(\dfrac{a\cos x}{\sqrt{a^2 + b^2}} + \dfrac{b\sin x}{\sqrt{a^2 + b^2}}\right)+c$

2. Now, say that there exists a right triangle whose sides are $a, b, \sqrt{a^2 + b^2}$ , such that;

$\sin\beta = \dfrac{a}{\sqrt{a^2 + b^2}};\quad \cos\beta = \dfrac{b}{\sqrt{a^2 + b^2}}$

3. Substitute the above values in the function to get:

$\begin{aligned} f(x)&=\sqrt{a^2 + b^2}\left(\sin\beta\cos x+ \cos\beta\sin x\right) + c\\ &=\sqrt{a^2 + b^2}\sin(\beta +x)+c\\ \end{aligned}$

4. Now let us say that the angle $\beta + x=\alpha$ , thus the maximum value of $\sin\alpha = 1$ and the minimum value of $\sin\alpha = -1$ , hence;

$\begin{aligned} f(x)_{\text{max}}&=\sqrt{a^2 + b^2}+c\\ f(x)_{\text{min}}&=-\sqrt{a^2 + b^2}+c\\ \end{aligned}$

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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Comments

That is a general method in practice. Anyway finding it on your own is great! !

Abhi Kumbale - 5 years ago

Oh I thought so, thanks!

Sravanth C. - 5 years ago

I try...we have $f(x)=a\cos x + b\sin x + c$ If we derive we obtain $f'(x)=-a\sin x + b\cos x$ To find max/min $-a\sin x + b\cos x=0$ We find $\cos x=\frac{a}{b}\sin x$ $\tan x=\frac{b}{a}$ So now we can replace them in f(x) $y_{max}=(\frac{a^2+b^2}{b})\sin(\tan^{-1}(\frac{b}{a})) + c$

but if we consider a rectangle triangle with sides a, b and c, where alpha is the angle between b and c (and the angle between a and b is 90°) then

$\sin(\tan^{-1}(\frac{b}{a}))=\frac{b}{c}=\frac{b}{\sqrt{a^2+b^2}}$

So now

$y_{max}=\sqrt{a^2+b^2} + c$

We can try to see that it is really the maximum of f(x), so the minimum symmetrically must be

$y_{min}=-\sqrt{a^2+b^2} + c$

Right...?

Matteo Monzali - 5 years ago

Exactly! I almost did the same! Splendid!

I have reduced some steps, thanks for posting it!

For an elegant proof try using Cauchy Schwarz Inequality.

Sal Gard - 5 years ago

@Matteo Monzali: Since you have posted this, I have given another version of what I did. Please check that out.

Yes, but I think your method is easier than mine! :)

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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$