/*! 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 6 Write each of the following in t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Write each of the following in terms of \(\sin \theta\) only. $$ \sec \theta $$

Short Answer

Expert verified
\(\sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}}\).

Step by step solution

01

Understand Secant in Terms of Cosine

The secant function, \(\sec \theta\), is the reciprocal of the cosine function. Therefore, by definition, \(\sec \theta = \frac{1}{\cos \theta}\).
02

Recall Pythagorean Identity

The Pythagorean identity states that \(\sin^2 \theta + \cos^2 \theta = 1\). We can use this identity to express \(\cos \theta\) in terms of \(\sin \theta\): \(\cos \theta = \sqrt{1 - \sin^2 \theta}\).
03

Substitute Cosine with Sine

Substitute \(\cos \theta\) in the expression for \(\sec \theta\): \(\sec \theta = \frac{1}{\sqrt{1 - \sin^2 \theta}}\).

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 is an important part of trigonometry. It helps us understand the relationship between angles in a right triangle and their ratios. The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. This means it is the reverse operation of the cosine function. Thus, we can say:
  • \( \sec \theta = \frac{1}{\cos \theta} \)
This expression shows that the secant function is directly linked to the cosine function. Remember, knowing the basic reciprocal trigonometric identities is essential in understanding how these functions relate to each other in various mathematical situations.
Cosine Function
The cosine function is one of the fundamental trigonometric functions. It relates to an angle in a right triangle by comparing the length of the adjacent side to the length of the hypotenuse. Here鈥檚 a quick summary of the cosine function:
  • \( \cos \theta \) represents the ratio of the adjacent side over the hypotenuse.
In terms of a unit circle, \( \cos \theta \) is the x-coordinate of a point on the circle. This function helps us get a clearer understanding of angles and geometric shapes.To link cosine with sine, we use the Pythagorean identity, which allows us to express \( \cos \theta \) in terms of \( \sin \theta \):
  • \( \cos \theta = \sqrt{1 - \sin^2 \theta} \)
With this relationship, you can convert expressions involving cosine into ones that only use sine.
Pythagorean Identity
The Pythagorean identity is a cornerstone in trigonometry. It's used to connect the sine and cosine functions through a simple yet powerful equation:
  • \( \sin^2 \theta + \cos^2 \theta = 1 \)
This identity is derived from the Pythagorean theorem and it plays a vital role in trigonometry. It helps transform expressions and solve trigonometric equations.Whenever you know the sine of an angle, you can find the cosine using this identity, and vice versa. Specifically, for our purpose of expressing everything in terms of \( \sin \theta \), we set it up as:
  • \( \cos \theta = \sqrt{1 - \sin^2 \theta} \)
By substituting into the secant function, this allows us to represent \( \sec \theta \) using just \( \sin \theta \), helping to streamline complex trigonometric problems.

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

Suppose the angle formed by the line \(y=3 x\) and the positive \(x\)-axis is \(\theta\). Find the tangent of \(\theta\) (Figure 1).

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ (\sin \theta-\cos \theta)^{2}-1=-2 \sin \theta \cos \theta $$

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\sin \theta\) if \(\cos \theta=\frac{12}{13}\) and \(\theta\) terminates in QI.

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\tan \theta\) if \(\sin \theta=\frac{1}{3}\) and \(\theta\) terminates in QI.

For Problems 23 through 26 , recall that \(\sin ^{2} \theta\) means \((\sin \theta)^{2}\). If \(\tan \theta=2\), find \(\tan ^{3} \theta\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.