Question

Amplitude and period Identify the amplitude and period of the following functions. $$g(\theta)=3 \cos (\theta / 3)$$

Short answer

Answer: The amplitude of the function is 3 and the period is \(6\pi\).

Step-by-step solution

Recognize the general form of the cosine function

The general form of the cosine function is given by: $$y = A \cos(Bx)$$ Where \(A\) is the amplitude, and the period is given by \(\dfrac{2\pi}{B}\). Our goal is to find the values of \(A\) and \(B\) in the given function, \( g(\theta) = 3 \cos (\theta / 3)\).

Identify the amplitude A

Comparing the given function \(g(\theta) = 3 \cos (\theta / 3)\) with the general form \(y = A \cos(Bx)\), we can see that the amplitude \(A\) is equal to the coefficient of the cosine function: $$A=3$$ So, the amplitude of the function is 3.

Identify the coefficient B

Again, comparing the given function \(g(\theta) = 3 \cos (\theta / 3)\) with the general form \(y = A \cos(Bx)\), we can see that the coefficient \(B\) is equal to the coefficient of the angle (inside the cosine function): $$B=1/3$$

Calculate the period

Now that we've found the value of \(B\), we can use the formula for the period of a trigonometric function: $$\text{period} = \dfrac{2\pi}{B}$$ In this case, the period is: $$\text{period} = \dfrac{2\pi}{1/3} = 6\pi$$ So, the period of the function is \(6\pi\).

State the amplitude and period

We have found the amplitude and period of the function \(g(\theta) = 3 \cos (\theta / 3)\). The amplitude is \(A = 3\), and the period is \(6\pi\).