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Chapter 7: Problem 21

Factor each trigonometric expression. $$\sin ^{3} x-\cos ^{3} x$$

Short Answer

Expert verified
The factorized form is \( ( \sin x - \cos x ) ( 1 + \sin x \cos x ) \.

Step by step solution

01

Identify the expression and apply the difference of cubes formula

Recognize that the given expression \( \sin ^{3} x-\cos ^{3} x \ \) is a difference of cubes. The difference of cubes formula is given by: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]Here, \( a = \sin x \ \) and \( b = \cos x \ \).
02

Substitute the values into the difference of cubes formula

Using \( a = \sin x \ \) and \( b = \cos x, \ \) rewrite the expression as: \( ( \sin x - \cos x ) ( \sin ^{2} x + \sin x \cos x + \cos ^{2} x ) \ \)
03

Simplify the second factor using trigonometric identities

Recall that \( \sin^2 x + \cos^2 x = 1. \ \) Use this identity to simplify the expression inside the second factor: \[ \sin^2 x + \sin x \cos x + \cos^2 x = 1 + \sin x \cos x \]Thus, the factorization is: \[ ( \sin x - \cos x ) ( 1 + \sin x \cos x ) \]

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Key Concepts

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

Difference of cubes
The difference of cubes formula is a helpful algebraic tool for factoring expressions of the form \( a^3 - b^3 \). Identifying expressions like \( \sin ^{3} x-\cos ^{3} x \) as a difference of cubes is the first step in simplification. The general formula is: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]. Here:
  • \( a = \sin x \)
  • \( b = \cos x \)
Substituting these values, the expression becomes \( ( \sin x - \cos x ) ( \sin ^{2} x + \sin x \cos x + \cos ^{2} x )\). This is the transformed version of the original trigonometric expression using the difference of cubes formula.
Trigonometric identities
Trigonometric identities are fundamental relationships between trigonometric functions. In this exercise, we use the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). This identity simplifies the expression inside the second factor. Initially, the second factor looks like \( \sin^2 x + \sin x \cos x + \cos^2 x \). By applying the Pythagorean identity:
  • \( \sin^2 x + \cos^2 x = 1 \)
we simplify it to \( 1 + \sin x \cos x \). Thus, the trigonometric identities allow us to transform and simplify complex trigonometric expressions effectively.
Trigonometric expressions
Trigonometric expressions involve functions like sine, cosine, and their combinations. Factoring these expressions requires familiarity with algebraic techniques like the difference of cubes, as well as trigonometric identities. In this case, we factored \( \sin ^{3} x-\cos ^{3} x \) using the difference of cubes formula. This resulted in \( ( \sin x - \cos x ) ( 1 + \sin x \cos x ) \). Knowing these techniques and identities makes solving trigonometric problems more manageable and allows for deeper understanding of the relationships within trigonometric functions.

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Most popular questions from this chapter

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval \([0,2 \pi)\). Express solutions to four decimal places. $$x^{3}-\cos ^{2} x=\frac{1}{2} x-1$$

Solve each problem. Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. Beats occur when two tones vary in frequency by only a few hertz. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of this slight variation in the frequency. This phenomenon can be seen using a graphing calculator. (Source: Pierce, \(\mathrm{J}\)., The Science of Musical Sound, Scientific American Books.) (a) Consider the two tones with frequencies of \(220 \mathrm{Hz}\) and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t\) and \(P_{2}=0.005\) sin \(446 \pi t,\) respectively. Graph the pressure \(P=P_{1}+P_{2}\) felt by an eardrum over the 1 -sec interval \([0.15,1.15] .\) How many beats are there in 1 sec? (b) Repeat part (a) with frequencies of 220 and \(216 \mathrm{Hz}\) (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.

Verify that each trigonometric equation is an identity. $$\frac{1-\cos \theta}{1+\cos \theta}=2 \csc ^{2} \theta-2 \csc \theta \cot \theta-1$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \([0,2 \pi)\) and your work leads to \(\frac{1}{2} x=\frac{\pi}{16}, \frac{5 \pi}{12}, \frac{5 \pi}{8} .\) What are the corresponding values of \(x ?\)

Verify that each equation is an identity. $$\cot 4 \theta=\frac{1-\tan ^{2} 2 \theta}{2 \tan 2 \theta}$$
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