[Math] If $y = a\sin{x} + b\cos{x} +C$ then find maxima and minima for $y$.

Question

I was able to solve it till

$$y = \sqrt{(a^2 + b^2)}\sin(\alpha + x) + C.$$

But I don't know how to find maxima and minima from here.

If $C = 0$ then maxima & minima equals the amplitude of the sine curve but when $C$ is non-zero then?

I need help from here onwards.

Answer 1

Hint:

• $\sin(\alpha +x) \in [-1,1]$

  $$ \therefore \quad \sqrt{a^2+b^2}\sin(\alpha+x) \in \left[ - \sqrt{a^2+b^2}, \sqrt{a^2+b^2} \right]$$

$$ \quad \therefore \sqrt{a^2+b^2}\sin(\alpha+x)+C \in \left[C- \sqrt{a^2+b^2}, C+ \sqrt{a^2+b^2}\ \right] \quad ,$$ therefore the maximum of $y$ is $_____$, and the minimum of $y$ is $_____$.