/*! 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 9 Write each expression in terms o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 9

Write each expression in terms of sines and/or cosines, and then simplify. \(\sec x \cos x\)

Short Answer

Expert verified
The simplified expression is \(1\).

Step by step solution

01

Understand the Given Expression

The given expression is \( \sec x \cos x \). Notice that \(\sec x\) is the secant function and it can be written in terms of cosine.
02

Rewrite Secant in Terms of Cosine

Recall that \( \sec x = \frac{1}{\cos x} \). Substitute this into the expression: \( \sec x \cos x = \left( \frac{1}{\cos x} \right) \cos x \).
03

Simplify the Expression

Now, simplify the expression: \( \left( \frac{1}{\cos x} \right) \cos x = \frac{\cos x}{\cos x} \). Since the numerator and denominator are the same, they cancel each other out, giving: \( 1 \).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

secant function
The secant function, written as \(\text{sec } x\), is one of the six fundamental trigonometric functions. It is closely related to the cosine function. The secant function is defined as the reciprocal of the cosine function. In mathematical terms, this is expressed as \[ \text{sec } x = \frac{1}{\text{cos } x} \].
Understanding how the secant function relates to the cosine function is essential. It allows you to rewrite expressions involving the secant function in terms of cosine. This can simplify complex trigonometric problems.
In the given exercise, we start with the expression \(\text{sec } x \text{ cos } x\). By substituting the reciprocal definition of the secant function \(\frac{1}{\text{cos } x}\), we can transform this into a more manageable form.
cosine function
The cosine function, denoted as \(\text{cos } x\), is fundamental in trigonometry. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The function is periodic with a period of \(\text{2}\text{蟺}\). It is essential to understand when working with trigonometric identities and simplifications.
In our exercise, we used the property of the cosine function while manipulating the expression \(\text{sec } x \text{ cos } x\). The cosine function's role becomes evident when we substitute \(\text{sec } x\) with \(\frac{1}{\text{cos } x}\), transforming the original expression into \[ \text{sec } x \text{ cos } x = \frac{1}{\text{cos } x} \text{ cos } x \].
This step simplifies the problem significantly, demonstrating how important it is to understand the properties of basic trigonometric functions.
trigonometric identities
Trigonometric identities are equations that are true for all values of the included variables. They are crucial tools in simplifying expressions and solving trigonometric equations. One of the fundamental identities is \[ \text{sec } x = \frac{1}{\text{cos } x} \].
In the exercise, we leveraged this identity to rewrite the expression \[ \text{sec } x \text{ cos } x = \frac{\text{cos } x}{\text{cos } x} = 1 \].
Other key trigonometric identities include:
  • \(\text{sin}^2 x + \text{cos}^2 x = 1\)
  • \(\text{tan } x = \frac{\text{sin } x}{\text{cos } x}\)
  • \(\text{csc } x = \frac{1}{\text{sin } x}\)
  • \(\text{cot } x = \frac{1}{\text{tan } x} = \frac{\text{cos } x}{\text{sin } x}\)
By mastering these identities, you can simplify and solve many trigonometric expressions efficiently.

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

For each equation, either prove that it is an identity or prove that it is not an identity. $$\tan \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos x}{1+\cos x}}$$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (\alpha / 2)=-1 / 3 \text { and } 7 \pi / 4<\alpha / 2<2 \pi$$

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$8 \cos ^{4} \theta-10 \cos ^{2} \theta+3=0$$

Find all values of \(\alpha\) in degrees that satisfy each equation. Round approximate answers to the nearest tenth of a degree. $$\sec 2 \alpha=4.5$$

Trigonometric Identities List as many trigonometric identities as you can and explain why each one is an identity.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.