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Chapter 7: Problem 22

Factor each trigonometric expression. $$\sin ^{3} \alpha+\cos ^{3} \alpha$$

Short Answer

Expert verified
(\text{\textbackslash sin} \text{\textbackslash alpha} + \text{\textbackslash cos} \text{\textbackslash alpha})(1 - \text{\textbackslash sin} \text{\textbackslash alpha} \text{\textbackslash cos} \text{\textbackslash alpha})

Step by step solution

01

Recognize the form of the expression

The given expression is \(\text{\textbackslash sin}^{3} \text{\textbackslash alpha} + \text{\textbackslash cos}^{3} \text{\textbackslash alpha}\). Notice it follows the form of a sum of cubes. Recall the sum of cubes formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \], where \( a = \text{\textbackslash sin} \text{\textbackslash alpha}\) and \( b = \text{\textbackslash cos} \text{\textbackslash alpha}\).
02

Apply the sum of cubes formula

Using the sum of cubes formula, rewrite the expression: \[ \text{\textbackslash sin}^3 \text{\textbackslash alpha} + \text{\textbackslash cos}^3 \text{\textbackslash alpha} = (\text{\textbackslash sin} \text{\textbackslash alpha} + \text{\textbackslash cos} \text{\textbackslash alpha})(\text{\textbackslash sin}^2 \text{\textbackslash alpha} - (\text{\textbackslash sin} \text{\textbackslash alpha})(\text{\textbackslash cos} \text{\textbackslash alpha}) + \text{\textbackslash cos}^2 \text{\textbackslash alpha}) \]
03

Simplify the second factor

Note that \( \text{\textbackslash sin}^2 \text{\textbackslash alpha} + \text{\textbackslash cos}^2 \text{\textbackslash alpha} = 1\), which is a fundamental trigonometric identity. Substitute this into our expression: \[ (\text{\textbackslash sin} \text{\textbackslash alpha} + \text{\textbackslash cos} \text{\textbackslash alpha})(\text{\textbackslash sin}^2 \text{\textbackslash alpha} - \text{\textbackslash sin} \text{\textbackslash alpha} \text{\textbackslash cos} \text{\textbackslash alpha} + 1 - \text{\textbackslash cos}^2 \text{\textbackslash alpha}) = (\text{\textbackslash sin} \text{\textbackslash alpha} + \text{\textbackslash cos} \text{\textbackslash alpha})(1 - \text{\textbackslash sin} \text{\textbackslash alpha}\text{\textbackslash cos} \text{\textbackslash alpha}) \]
04

Combine and simplify

Thus, our factored expression is: \[ \text{\textbackslash sin}^{3} \text{\textbackslash alpha}+\text{\textbackslash cos}^{3} \text{\textbackslash alpha} = (\text{\textbackslash sin} \text{\textbackslash alpha} + \text{\textbackslash cos} \text{\textbackslash alpha})(1 - \text{\textbackslash sin} \text{\textbackslash alpha} \text{\textbackslash cos} \text{\textbackslash alpha}) \]

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Key Concepts

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

sum of cubes
The given trigonometric expression \(\text{\textbackslash sin}^{3} \text{\textbackslash alpha} + \text{\textbackslash cos}^{3} \text{\textbackslash alpha}\) follows the form of a sum of cubes. The sum of cubes formula states that: \[ a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) \]In our case, \( a = \text{\textbackslash sin} \text{\textbackslash alpha} \) and \( b = \text{\textbackslash cos} \text{\textbackslash alpha} \). By recognizing this, we can then rewrite and factor the expression accordingly. Identifying which form to use鈥攚hether it's the sum of cubes, difference of cubes, or another polynomial identity鈥攊s a crucial skill in algebra and greatly simplifies the problem.
trigonometric identities
Trigonometric identities are equations that are true for all values of the included variables. They help simplify and solve trigonometric problems. One fundamental identity we use here is:\[ \text{sin}^{2} \text{\textbackslash alpha} + \text{cos}^{2} \text{\textbackslash alpha} = 1 \]When we applied the sum of cubes formula and expanded the factors, this identity was crucial in simplifying the expression further. By substituting \( \text{sin}^{2} \text{\textbackslash alpha} + \text{\textbackslash cos}^{2} \text{\textbackslash alpha} = 1 \), we simplified the second factor significantly. Understanding and memorizing these identities is essential for solving trigonometric equations efficiently.
simplification steps
Simplifying trigonometric expressions involves several steps, as shown in the provided solution.
  • Firstly, recognize the form or pattern of the expression. In this case, it was a sum of cubes.
  • Secondly, apply the relevant formula or identity. Here, we used the sum of cubes formula.
  • Thirdly, simplify each part of the expression using trigonometric identities, like \( \text{\textbackslash sin}^{2} \text{\textbackslash alpha} + \text{\textbackslash cos}^{2} \text{\textbackslash alpha} = 1 \).
  • Lastly, combine and simplify all parts to get the final factored expression.
Following a systematic approach helps break down complex problems into manageable steps. Practice these steps, and soon they will become second nature!

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Most popular questions from this chapter

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 \(x ?\)

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(\frac{1}{3} \theta=45^{\circ}, 60^{\circ}, 75^{\circ}, 90^{\circ} .\) What are the corresponding values of \(\theta ?\)

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(3 \theta=180^{\circ}, 630^{\circ}, 720^{\circ}, 930^{\circ} .\) What are the cor- responding values of \(\theta ?\)

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.

The study of alternating electric current requires the solutions of equations of the form $$ i=I_{\max } \sin 2 \pi f t $$ for time t in seconds, where is instantaneous current in amperes. \(I_{\max }\) is maximum cur. rent in amperes, and F is the number of cycles per second.Find the least positive value of \(t,\) given the following data. $$i=50, I_{\max }=100, f=120$$
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