Question

determine the phase shift (in rad) of the sinusoidal function ( g(x)=-3cos(pi-\frac{x}{10}) ).

( pi ) right

( 10pi ) right

( pi ) left

( 10 ) left

Explanation:

Step1: Rewrite the function

Use the identity \( \cos(A - B)=\cos A\cos B+\sin A\sin B\). But a better way is to use the formula for phase - shift of \(y = A\cos(Bx - C)+D\). The phase - shift is given by \(\frac{C}{B}\). First, rewrite \(g(x)=-3\cos(\pi-\frac{x}{10})\) as \(g(x)=-3\cos(-(\frac{x}{10}-\pi))\). Since \(\cos(-\theta)=\cos\theta\), then \(g(x)=-3\cos(\frac{x}{10}-\pi)\). Here \(A=-3\), \(B = \frac{1}{10}\), \(C=\pi\).

Step2: Calculate the phase - shift

The formula for the phase - shift of the function \(y = A\cos(Bx - C)+D\) is \(\text{Phase - shift}=\frac{C}{B}\). Substitute \(C = \pi\) and \(B=\frac{1}{10}\) into the formula. We get \(\text{Phase - shift}=\frac{\pi}{\frac{1}{10}}=10\pi\). Since the function is of the form \(y=\cos(Bx - C)\) (\(Bx-C=\frac{x}{10}-\pi\)), and when \(Bx - C=0\) (for the starting point of the cosine wave), \(x = 10\pi\). For \(y = \cos(x)\), the starting point is \(x = 0\). So the graph of \(y=-3\cos(\frac{x}{10}-\pi)\) is the graph of \(y = \cos(x)\) shifted \(10\pi\) units to the right.

Answer:

\(10\pi\) right