How do you find the amplitude, period, phase shift for y=sin(x+pi/3)?

Apr 8, 2018

See the explanation below.

Explanation:

The sine function is defined for

$\forall x \in \mathbb{R}$,

$- 1 \le \sin x \le 1$

The amplitude $= 1$

To calculate the period of a T-periodic function

Let $f \left(x\right) = \sin \left(x + \frac{\pi}{3}\right)$

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

$f \left(x + T\right) = \sin \left(x + \frac{\pi}{3} + T\right) = \sin \left(x + \frac{\pi}{3}\right) \cos T + \cos \left(x + \frac{\pi}{3}\right) \sin T$

$\sin \left(x + \frac{\pi}{3}\right) \cos T + \cos \left(x + \frac{\pi}{3}\right) \sin T = \sin \left(x + \frac{\pi}{3}\right)$

$\cos T = 1$ and $\sin T = 0$

$T = 0 , T = 2 \pi$, $\left[2 \pi\right]$

The period is $T = 2 \pi$

The phase shift is $= \frac{\pi}{3}$

graph{sin(x+pi/3) [-8.23, 17.08, -6.37, 6.29]}