Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 43

Chapter 2, Problem 43

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = cos(x − π/2)

Verified step by step guidance

Identify the general form of the cosine function: \(y = A \cos(B(x - C))\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) is the phase shift.

Compare the given function \(y = \cos(x - \frac{\pi}{2})\) to the general form. Here, \(A = 1\) (since there is no coefficient in front of cosine), \(B = 1\) (coefficient of \(x\)), and \(C = \frac{\pi}{2}\).

Calculate the amplitude, which is the absolute value of \(A\): \(\text{Amplitude} = |A| = 1\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Since \(B = 1\), the period is \(2\pi\).

Determine the phase shift, which is \(C\). Since the function is \(\cos(x - \frac{\pi}{2})\), the phase shift is \(\frac{\pi}{2}\) units to the right.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a trigonometric function from its midline. For functions like y = cos(x), the amplitude is the coefficient before the cosine term, indicating the height of peaks and depth of troughs. In y = cos(x − π/2), the amplitude is 1, as there is no coefficient other than 1.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function, typically 2π for sine and cosine functions. It can be adjusted by a coefficient inside the function's argument. For y = cos(x − π/2), since the coefficient of x is 1, the period remains 2π.

Phase Shift of a Trigonometric Function

Phase shift refers to the horizontal translation of the graph, determined by the value added or subtracted inside the function's argument. For y = cos(x − π/2), the graph shifts π/2 units to the right, meaning the entire cosine curve moves rightward by π/2 along the x-axis.

Recommended video:

6:31

Phase Shifts

Related Practice

Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec(x + π)

views

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

views

Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)

Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = cos(x + π/2)

Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = csc(x − π)