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Chapter 3: Problem 28

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

Short Answer

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

Step by step solution

01

Identify the Trigonometric Expression

Recognize that we are given the expression to factor: \[9 \csc ^{2} (\theta) - 12 \csc (\theta) + 4\]
02

Substitute a Variable

To simplify the process, use a substitution. Let \[ u = \csc (\theta) \]. Now the expression can be rewritten as: \[ 9u^{2} - 12u + 4\]
03

Factor the Quadratic Expression

Factor the quadratic expression \[ 9u^{2} - 12u + 4\]. Notice it can be factored as: \[ (3u - 2)(3u - 2) \] or \[ (3u - 2)^{2} \]
04

Substitute Back the Original Trigonometric Function

Replace \[ u \] with \[ \csc (\theta) \]. This gives: \[ (3 \csc (\theta) - 2)^{2} \]

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. These identities help simplify trigonometric expressions and solve trigonometric equations.
Understanding these identities is essential because they form the foundation of trigonometry.
Some common trigonometric identities include:
  • Pythagorean Identities: \(\text{sin}^{2}(胃) + \text{cos}^{2}(胃) = 1 \)
  • Reciprocal Identities: \(\text{csc}(胃) = \frac{1}{\text{sin}(胃)} \)
  • Quotient Identities: \(\text{tan}(胃) = \frac{\text{sin}(胃)}{\text{cos}(胃)}\)
Recognizing these identities makes it easier to manipulate and factor expressions, just like in the given exercise where \(\text{csc}(胃)\) was used.
Substitution Method in Trigonometry
The substitution method in trigonometry involves replacing a trigonometric function with a single variable to simplify calculations.
For example: In the given exercise, the substitution \(\text{u} = \text{csc}(胃)\) was used.
This transformation allowed us to rewrite the complex trigonometric expression as a simpler quadratic equation: \(9u^{2} - 12u + 4\).
This substitution streamlines the process of factoring and solving trigonometric equations by temporarily converting them into algebraic ones.
After simplifying, we revert back to the original trigonometric functions.
Factoring Quadratic Expressions
Factoring quadratic expressions is the process of breaking down a quadratic equation into a product of binomials.
In algebra, this can be represented as: \((ax^{2} + bx + c) = (px + q)(rx + s)\).
For example, in the exercise, we factored the quadratic expression \((9u^{2} - 12u + 4)\) into \((3u - 2)(3u - 2)\) or \((3u - 2)^{2}\).
This concept is vital in solving quadratic equations and simplifying expressions.
The key steps for factoring include:
  • Identifying the quadratic form
  • Recognizing patterns (like perfect squares)
  • Using the distributive property to check the factorization
With practice, recognizing these patterns and applying the factoring techniques becomes much easier.

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Most popular questions from this chapter

Find the exact value of \(\cos (\alpha+\beta)\) if \(\sin \alpha=2 / 3\) and \(\sin \beta=-1 / 2,\) with \(\alpha\) in quadrant \(I\) and \(\beta\) in quadrant III.

Use identities to simplify each expression. Do not use a calculator. \(2 \sin \left(\frac{\pi}{9}-\frac{\pi}{2}\right) \cos \left(\frac{\pi}{2}-\frac{\pi}{9}\right)\)

The equation \(f_{1}(x)=f_{2}(x)\) is an identity if and only if the graphs of \(y=f_{1}(x)\) and \(y=f_{2}(x)\) coincide at all values of \(x\) for which both sides are defined. Graph \(y=f_{1}(x)\) and \(y=f_{2}(x)\) on the same screen of your calculator for each of the following equations. From the graphs, make a conjecture as to whether each equation is an identity, then prove your conjecture. $$ \tan x+\sec x=\frac{\sin ^{2} x+1}{\cos x} $$

Find the point that lies midway between \((\pi / 3,1)\) and \((\pi / 2,1)\)

Find the exact value of \(\cos (\alpha-\beta)\) if \(\cos \alpha=\sqrt{3} / 4\) and \(\cos \beta=-\sqrt{2} / 3,\) with \(\alpha\) in quadrant \(I\) and \(\beta\) in quadrant II.
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