/*! 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 Simplify. Check your results usi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 32

Simplify. Check your results using a graphing calculator. $$\cos ^{4} x-\sin ^{4} x$$

Short Answer

Expert verified
\(\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x\).

Step by step solution

01

- Recognize the Algebraic Identity

We need to recognize that the expression is in the form of a difference of squares. The difference of squares formula is given by \[a^2 - b^2 = (a - b)(a + b)\].
02

- Apply the Identity

Set \(a = \cos^2 x\) and \(b = \sin^2 x\). Substituting these variables into the difference of squares formula, we get: \[\cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\].
03

- Simplify Using Pythagorean Identity

Recall that \(\cos^2 x + \sin^2 x = 1\), which is the Pythagorean identity. Substituting this into our equation, we have: \[(\cos^2 x - \sin^2 x)(1) = \cos^2 x - \sin^2 x \].
04

- Confirm via Graphing Calculator

Enter the original expression \(\cos^4 x - \sin^4 x\) into the graphing calculator and compare it to \(\cos^2 x - \sin^2 x\). Both graphs should be identical, confirming our simplification.

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
In algebra, the difference of squares is a special pattern you can use to simplify expressions. It states that for any two numbers or expressions, say \(a\) and \(b\), their difference squared can be expressed as: \[a^2 - b^2 = (a - b)(a + b)\]
This pattern is very useful for breaking down complex expressions and simplifying them. In our exercise, we used this identity by setting \(a = \cos^2 x\) and \(b = \sin^2 x\). As a result, our expression \( \cos^4 x - \sin^4 x\) was simplified into: \[(\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\].
Remember, recognizing these patterns can make solving problems much easier!
Pythagorean Identity
The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that: \[\cos^2 x + \sin^2 x = 1\] This is derived from the Pythagorean theorem applied to a right triangle.
In the context of our exercise, after applying the difference of squares, we had the expression: \[ (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\]
Using the Pythagorean identity, we know that \(\cos^2 x + \sin^2 x\) is equal to 1. This simplified our expression further to just: \(\cos^2 x - \sin^2 x\).
Recognizing and applying the Pythagorean identity quickly simplifies expressions involving trigonometric functions.
Graphing Calculator Validation
Graphing calculators can be extremely helpful tools for validating your work. After simplifying an expression, you can check your result by graphing both the original and simplified expressions.
In our exercise, we compared the graphs of \( \cos^4 x - \sin^4 x\) and \( \cos^2 x - \sin^2 x\). Both graphs should overlap perfectly if our simplification is correct.
To do this:
  • Input the original expression \( \cos^4 x - \sin^4 x\) into the calculator.
  • Graph it.
  • Then, input the simplified expression \(\cos^2 x - \sin^2 x\).
  • Graph it and compare.

Using a graphing calculator in this way not only confirms your work but also strengthens your understanding of how algebraic and trigonometric manipulations affect functions visually.

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

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ Trigonometric functions with domains composed of real numbers are called ____________ functions.

Line \(l_{1}\) contains the points \((-3,7)\) and \((-3,-2)\) Line \(l_{2}\) contains \((0,-4)\) and \((2,6) .\) Find the smallest positive angle from \(l_{1}\) to \(l_{2}\)

Solve. $$x-7=x+3.4$$

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a period of \(\pi ?\)

Assuming that \(\sin \theta=0.6249\) and \(\cos \phi=0.1102\) and that both \(\theta\) and \(\phi\) are first-quadrant angles, evaluate each of the following. $$\sin (\theta-\phi)$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision 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.