# How do you find the amplitude and period of y=cos2x?

May 28, 2018

The period is $T = \pi$.
The amplitude is $= 1$

#### Explanation:

We need

$\cos \left(a + b\right) = \cos a \cos b - \sin a \sin b$

The period of a periodic function is $T$ iif

$f \left(x\right) = f \left(x + T\right)$

Here,

$f \left(x\right) = \cos \left(2 x\right)$

Therefore,

$f \left(x + T\right) = \cos \left(2 \left(x + T\right)\right)$

where the period is $= T$

So,

$\cos \left(2 x\right) = \cos \left(2 \left(x + T\right)\right) = \cos \left(2 x + 2 T\right) = \cos 2 x \cos 2 T - \sin 2 x \sin 2 T$

$\implies$, $\left\{\begin{matrix}\cos 2 T = 1 \\ \sin 2 T = 0\end{matrix}\right.$

$\implies$, $2 T = 2 \pi$

$\implies$, $T = \pi$

As

$- 1 \le \cos x \le 1$

Therefore,

$- 1 \le \cos \left(2 x\right) \le 1$

The amplitude is $= 1$

graph{cos(2x) [-1.322, 9.777, -2.333, 3.214]}