Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.3.38

Chapter 1, Problem 1.3.38

What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

Verified step by step guidance

Consider the addition formulas for sine and cosine: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) and \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).

Substitute \( B = 2\pi \) into these formulas. Recall that \( \cos(2\pi) = 1 \) and \( \sin(2\pi) = 0 \).

For the sine addition formula, substitute \( B = 2\pi \): \( \sin(A + 2\pi) = \sin A \cdot 1 + \cos A \cdot 0 = \sin A \).

For the cosine addition formula, substitute \( B = 2\pi \): \( \cos(A + 2\pi) = \cos A \cdot 1 - \sin A \cdot 0 = \cos A \).

These results show that \( \sin(A + 2\pi) = \sin A \) and \( \cos(A + 2\pi) = \cos A \), which agree with the periodicity of sine and cosine functions, confirming that they repeat every \( 2\pi \).

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.

Addition Formulas

Addition formulas in trigonometry express the sine and cosine of the sum or difference of two angles. For example, the sine addition formula states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas are essential for simplifying expressions involving trigonometric functions and for solving equations that include angles.

Periodic Functions

Trigonometric functions such as sine and cosine are periodic, meaning they repeat their values in regular intervals. The period of sine and cosine is 2π, which indicates that sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x). Understanding periodicity is crucial when analyzing the behavior of trigonometric functions over different intervals.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of trigonometric functions, where the x-coordinate represents the cosine and the y-coordinate represents the sine of an angle. Using the unit circle helps visualize how angles relate to their sine and cosine values, especially when considering angles like B = 2π.

Recommended video:

5:10

Evaluate Composite Functions - Values on Unit Circle

Related Practice

Textbook Question

Composition of Functions

A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.

Textbook Question

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

cos (x − π/2) = sin x

Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph the function f (x) = sin³ x.

Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.

Graph four periods of the function f (x) = −tan 2x.

Textbook Question

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = 3 cos 60x

Textbook Question

Can a function be both even and odd? Give reasons for your answer.