what is the maximum/minimum value of f(x)= ab $ \sin x + b \sqrt(1-a^2)\cos x + c$

Question

1)What is the maximum/minimum value of f(x) = ab $ \sin x + b \sqrt(1-a^2)\cos x$ + c? $|a| \lt 1, b \gt 0$ . 2) Find the maximum if c = 0 3) Find x if f(x) = c I tried trignometric substitution but alas! Can you please help? Thanks.

Answer 1

Let $a = \sin\theta$.

Then the above equation becomes $b\sin\theta\sin{x}+b\cos\theta\cos{x} +c $ or

$b\cos(\theta-x)+c$.

The maximum value obtained when $\theta-x=0$ in that case maximum value is $b+c$.

Also minimum happens when $\theta-x=\pi$ in that case minimum value is $-b+c$

$f(x) = c$ means, $b\cos(\theta-x)=0$ or $\cos(\theta-x)=0$ or,

$x = \theta- \pi/2$, or $x = \theta+ \pi/2$