/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 66 Use the unit circle to verify th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 66

Use the unit circle to verify that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd.

Short Answer

Expert verified
Cosine and Secant functions are even; and Sine, Cosecant, Tangent, and Cotangent functions are odd.

Step by step solution

01

Identifying Even Functions

Step 1: Start with the cosine function. Cos(胃) on the unit circle is defined as the x-coordinate of the point, and Cos(-胃) is the x-coordinate of the opposite point -- which is the same. Therefore, Cos(-胃) = Cos(胃), so Cos is an even function.\nNext, consider the secant function, Sec(胃) = 1/Cos(胃), is an even function because it is the reciprocal of an even function.
02

Identifying Odd Functions

Step 2: Now, consider sine function, Sin(胃) on the unit circle is defined as the y-coordinate of the point, and Sin(-胃) is the y-coordinate of the opposite point, which is the same. So, Sin(-胃) = -Sin(胃), indicating Sin is an odd function. The cosecant function Csc(胃) = 1/Sin(胃) is odd because it is the reciprocal of an odd function.\nTangent function, Tan(胃) = Sin(胃)/Cos(胃), and Cotangent function, Cot(胃) = Cos(胃)/Sin(胃), are both odd functions since the numerator is an odd function and the denominator is an even function, and the quotient of an odd function and an even function is odd.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ g(x)=\csc x $$

Evaluate the expression without using a calculator. $$ \arctan \sqrt{3} $$

Use a graphing utility to graph the function. Describe the behavior of the function as \(x\) approaches zero. $$ y=\frac{4}{x}+\sin 2 x, \quad x>0 $$

Use a graphing utility to graph the function. Include two full periods. $$ y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right) $$

Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\sec x \approx 1+\frac{x^{2}}{2 !}+\frac{5 x^{4}}{4 !}$$ where \(x\) is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.