# What is the amplitude and period of x=asin(nt)+bcos(nt)?

Apr 6, 2017

Amplitude is $\sqrt{{a}^{2} + {b}^{2}}$ and period is $\frac{2 \pi}{n}$.

#### Explanation:

As $x = a \sin \left(n t\right) + b \cos \left(n t\right)$, we can write ir as $x = b \cos \left(n t\right) + a \sin \left(n t\right)$

Now let $\cos \alpha = \frac{a}{\sqrt{{a}^{2} + {b}^{2}}}$ and $\sin \alpha = \frac{b}{\sqrt{{a}^{2} + {b}^{2}}}$

Observe that as $| \frac{a}{\sqrt{{a}^{2} + {b}^{2}}} | < 1$ and $| \frac{b}{\sqrt{{a}^{2} + {b}^{2}}} | < 1$, it is possible as ${\cos}^{2} \alpha + {\sin}^{2} \alpha = 1$

Then $x = \sqrt{{a}^{2} + {b}^{2}} \left[\cos \left(n t\right) \cos \alpha + \sin \left(n t\right) \sin \alpha\right]$

= $\sqrt{{a}^{2} + {b}^{2}} \cos \left(n t - \alpha\right)$

Hence amplitude is $\sqrt{{a}^{2} + {b}^{2}}$ and period is $\frac{2 \pi}{n}$.