/*! 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 30 Factor each trigonometric expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 30

Factor each trigonometric expression. \(\sin ^{4} y-\cos ^{4} x\)

Short Answer

Expert verified
(sin^2(y) + cos^2(x))(sin^2(y) - cos^2(x))

Step by step solution

01

- Recognize the structure

Observe that the expression \(\text{sin}^4(y) - \text{cos}^4(x)\) resembles the difference of squares form \(a^2 - b^2 = (a+b)(a-b)\). In this case, let \(a = \text{sin}^2(y)\) and \(b = \text{cos}^2(x)\).
02

- Apply the difference of squares

Using the difference of squares formula, rewrite the expression as: \((\text{sin}^2(y) + \text{cos}^2(x)) (\text{sin}^2(y) - \text{cos}^2(x))\).
03

- Simplify using Pythagorean identity

Recalling the Pythagorean identity \(\text{sin}^2(y) + \text{cos}^2(y) = 1\), we see that this does not directly simplify \( \text{sin}^2(y) + \text{cos}^2(x)\). Therefore, the result of Step 2 remains unchanged in one of the factors.
04

- Recognize the other factor

The factor \(\text{sin}^2(y) - \text{cos}^2(x)\) cannot be simplified further without additional information about the relationship between \(x\) and \(y\). Therefore, the complete factored form is: \( (\text{sin}^2(y) + \text{cos}^2(x))(\text{sin}^2(y) - \text{cos}^2(x)) \).

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.

difference of squares
The difference of squares is a special algebraic identity and an incredibly useful factorization technique. When you have an expression in the form of \(a^2 - b^2\), it can be factored into two simpler expressions: \((a + b)(a - b)\). This identity helps simplify complex expressions and solve equations faster.

In our example, the expression \( \text{sin}^4(y) - \text{cos}^4(x) \) fits the difference of squares form. We can set:
  • a = \( \text{sin}^2(y) \)
  • b = \( \text{cos}^2(x) \)

Applying the difference of squares formula: \( (\text{sin}^2(y) + \text{cos}^2(x)) (\text{sin}^2(y) - \text{cos}^2(x)) \). The expression is now factored and more manageable.
trigonometric identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are essential for simplifying expressions and solving trigonometric equations. The main trigonometric identities include:
  • Pythagorean identities
  • Reciprocal identities
  • Quotient identities

In our factoring problem, the identity we mainly utilize is the Pythagorean identity. Although other identities can come in handy, recognizing and applying the right one is crucial.
Pythagorean identity
The Pythagorean identity is one of the most fundamental trigonometric identities. It states that: \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1 \).

It's derived from the Pythagorean theorem in the context of the unit circle. For any angle \( \theta \), the sum of the square of its sine and cosine is always one.

When factoring our expression \( \text{sin}^4(y) - \text{cos}^4(x) \), the term \( \text{sin}^2(y) + \text{cos}^2(x) \) might resemble the Pythagorean identity. However, it does not simplify to one since it involves different variables (y and x). Therefore, understanding the limitations and proper application of this identity keeps our work accurate.

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

Prove that each of the following equations is an identity. HINT \(\ln (a / b)=\ln (a)-\ln (b)\) and \(\ln (a b)=\ln (a)+\ln (b)\) for \(a>0\) and \(b>0\) $$ \ln |\csc \alpha+\cot \alpha|=-\ln |\csc \alpha-\cot \alpha| $$

Find the exact value of \(\cos (\alpha+\beta)\) if \(\sin \alpha=-7 / 25\) and \(\sin \beta=8 / 17,\) with \(\alpha\) in quadrant IV and \(\beta\) in quadrant II.

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. $$ \frac{1}{1-\sin x}+\frac{1}{1+\sin x}=\frac{2}{\cos ^{2} x} $$

52\. \(\frac{1-\cos ^{2}\left(\frac{x}{2}\right)}{1-\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1-\cos x}{1+\cos x}\)

Show that \(\cos (\alpha-\beta)=\cos \alpha-\cos \beta\) is not an identity.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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.