Problem 44 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\cos \left(x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The amplitude of the function is 1. The period of the function is \(2\pi\). The phase shift is \(-\frac{\pi}{2}\). The graph starts at \(-\frac{\pi}{2}\), and ends at \(\frac{3\pi}{2}\), completing one whole period.

Step 1: Identify the amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient of the function. In the function \(y=\cos\left(x+\frac{\pi}{2}\right)\), the coefficient is 1 (since it's not explicitly stated). Therefore, the amplitude of the function is \(|1|=1\).

Step 2: Determine the period

The period of a standard cosine function is \(2\pi\). In the given function, there is no coefficient to \(x\), so the period is not affected and remains \(2\pi\).

Step 3: Identify the phase shift

The phase shift of a function is determined by the value added or subtracted from \(x\) inside the function. For the function \(y=\cos\left(x+\frac{\pi}{2}\right)\), the value added is \(\frac{\pi}{2}\), which means there is a phase shift to the left by \(\frac{\pi}{2}\).

Step 4: Graph the function

The graph of one period of the function would start at the point corresponding to \(-\frac{\pi}{2}\) (due to the phase shift), rise to 1 (the amplitude), fall back to -1, and then rise back to 1, completing the cycle at the point \(2\pi-\frac{\pi}{2} = \frac{3\pi}{2}\).

Step 5: Repeat the cycle

Finally, to complete the graph for one period, the same cycle will be repeated starting from \(\frac{3\pi}{2}\).