/*! 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 32 Factor each trigonometric expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 32

Factor each trigonometric expression. $$9 \csc ^{2} \theta-12 \csc \theta+4$$

Short Answer

Expert verified
(3 \text{csc} \theta - 2)^2

Step by step solution

01

Recognize the Trigonometric Identity

Identify that the given expression is a quadratic expression in terms of \( \text{csc} \theta \). In this case, \( 9 \text{csc}^{2} \theta - 12 \text{csc} \theta + 4 \).
02

Substitute a Single Variable

To simplify, let \( u = \text{csc} \theta \). The expression becomes \( 9u^{2} - 12u + 4 \).
03

Factor the Quadratic Equation

Factor the quadratic expression \( 9u^2 - 12u + 4 \) by finding two binomials that multiply to give the original quadratic. This results in \( (3u - 2)(3u - 2) \). We can verify by expanding: \ \[ (3u - 2)(3u - 2) = 9u^2 - 6u - 6u + 4 = 9u^2 - 12u + 4 \].
04

Substitute Back the Trigonometric Function

Replace \( u \) with \( \text{csc} \theta \). Thus, the factored form of the original expression is \[ (3 \text{csc} \theta - 2)(3 \text{csc} \theta - 2) \] or \[ (3 \text{csc} \theta - 2)^2 \].

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.

trigonometric identity
Trigonometric identities are important tools in solving trigonometric equations. They are equations that are true for all values of the variable involved. One common identity is \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \). In the given problem, we look at \( 9 \text{csc}^2 \theta - 12 \text{csc} \theta + 4 \) and can identify it as a quadratic expression using the identity \( \text{csc} \theta \).
quadratic equation
A quadratic equation is a second-degree polynomial equation in the form of \( ax^2 + bx + c = 0 \). In this case, by identifying \( u = \text{csc} \theta \), we transform our trigonometric expression into a quadratic form: \( 9u^2 - 12u + 4 \). This makes it easier to handle and solve.
substitution method
The substitution method simplifies complex expressions by replacing one variable with another. By setting \( u = \text{csc} \theta \), we can convert the trigonometric expression \( 9 \text{csc}^2 \theta - 12 \text{csc} \theta + 4 \) into the quadratic form \( 9u^2 - 12u + 4 \). This makes it easier to apply standard algebraic techniques for solving or factoring equations.
factoring binomials
Factoring binomials involves expressing a polynomial as a product of simpler binomial expressions. For the quadratic \( 9u^2 - 12u + 4 \), we look for two numbers that multiply to give the constant term (4) and add up to the coefficient of the linear term (-12). This results in \( (3u - 2)(3u - 2) \). Substituting back \( u = \text{csc} \theta \), we get \( (3 \text{csc} \theta - 2)(3 \text{csc} \theta - 2) \) or \( (3 \text{csc} \theta - 2)^2 \). This is the factored form of the original expression.

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

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. $$8 \cos ^{4} \theta-10 \cos ^{2} \theta+3=0$$

Prove that each equation is an identity. $$\cos ^{2}(A-B)-\cos ^{2}(A+B)=\sin ^{2}(A+B)-\sin ^{2}(A-B)$$

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. $$3 \sin 2 \theta=\cos 2 \theta$$

Complete the sum and difference identities. a. \(\cos (x+y)=\) ______ b. \(\cos (x-y)=\) ______

Use identities to simplify each expression. \(\csc ^{4} x-\cot ^{4} x\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.