# What is the maximum value of 5sin(t) + 12cos(t)?

May 6, 2016

The maximum is 13 and minimum is $- 13$.

#### Explanation:

$f \left(t\right) = 5 \sin t + 12 \cos t$. We can compound these two oscillations into a single sine oscillation.

$f \left(t\right) = 13 \left(\left(\frac{5}{13}\right) \sin t + \left(\frac{12}{13}\right) \cos t\right)$

$= 13 \left(\cos b \sin t + \sin b \cos t\right)$

$= 13 \sin \left(t + b\right)$, where $\sin b = \frac{12}{13} \mathmr{and} \cos b = \frac{5}{13}$.

So, $- 13 \le f \left(t\right) = 13 \sin \left(t + b\right) \le 13$

The amplitude of the oscillation is 13 and the period is $2 \pi$..