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Chapter 7: Problem 36

Write each expression as a sum or difference of trigonometric functions. $$5 \cos 3 x \cos 2 x$$

Short Answer

Expert verified
The expression as a sum or difference is \( \frac{5}{2} [\cos 5x + \cos x] \).

Step by step solution

01

Identify the Trigonometric Identity

To rewrite the expression as a sum or difference, use the product-to-sum identities for cosine. Specifically, the identity for cosine of products is: \[ \cos A \cos B = \frac{1}{2} [\cos (A+B) + \cos (A-B)] \]
02

Apply the Identity

In the given expression, let \( A = 3x \) and \( B = 2x \). Therefore, replace \( A \) and \( B \) in the identity: \[ \cos 3x \cos 2x = \frac{1}{2} [\cos (3x + 2x) + \cos (3x - 2x)] \]
03

Simplify the Expression

Now simplify the terms inside the cosine functions: \[ \cos 3x \cos 2x = \frac{1}{2} [\cos 5x + \cos x] \]
04

Multiply by the Constant

Finally, multiply the entire expression by the constant 5: \[ 5 \cos 3x \cos 2x = 5 \cdot \frac{1}{2} [\cos 5x + \cos x] = \frac{5}{2} [\cos 5x + \cos x] \]

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Key Concepts

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

Product-to-Sum Identities
Product-to-sum identities help simplify products of trigonometric functions into sums or differences. This transformation is useful in integration, simplifying expressions, and solving trigonometric equations. One essential product-to-sum identity is:
\[ \frac{1}{2} [\text{cos}(A+B) + \text{cos}(A-B)] \] for the product of two cosine functions.
In the given problem, we use this identity to transform \[ \text{cos}(3x) \text{cos}(2x) \].
Understanding these patterns is crucial, as they frequently appear in calculus and physics.
Cosine Function
The cosine function, represented as \[ \text{cos}(x) \], is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right triangle.
The cosine function has several key properties:
  • Range: \[-1, 1 \]
  • Periodicity: \[2\text{蟺} \]
  • Even Function: \[\text{cos}(-x) = \text{cos}(x) \]
In the context of this exercise, the cosine function transformation utilizes properties including sum and difference identities.
Trigonometric Expressions
Trigonometric expressions combine various trigonometric functions. Simplifying these expressions often involves using identities such as product-to-sum or Pythagorean identities.
The expression \[ 5 \text{cos}(3x) \text{cos}(2x) \] in this problem showcases the process. We begin by identifying and applying a suitable identity, transforming the product into sums.
  • Identification of functions and angles: \[A = 3x \text{ and } B = 2x \]
  • Application of product-to-sum identity: \[ \frac{1}{2} [\text{cos}(5x) + \text{cos}(x)] \]
  • Final multiplication by the constant: \[ \frac{5}{2} [\text{cos}(5x) + \text{cos}(x)] \]
Mastery of these steps allows students to tackle more complex trigonometric problems confidently.

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$3 \csc ^{2} \frac{x}{2}=2 \sec x$$

Solve each problem. When a musical instrument creates a tone of \(110 \mathrm{Hz}\). it also creates tones at \(220,330,440,550,660, \ldots\) Hz. A small speaker cannot reproduce the \(110-\mathrm{Hz}\) vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is \(110 \mathrm{Hz}\), and this difference tone will be heard by a listener. (Source: Benade, A.. Fundamentals of Musical Acoustics, Dover Publications.) (a) We can model this phenomenon using a graphing calculator. In the window \([0,0.03]\) by \([-1,1],\) graph the upper harmonics represented by the pressure $$ P=\frac{1}{2} \sin [2 \pi(220) t]+\frac{1}{3} \sin [2 \pi(330) t]+\frac{1}{4} \sin [2 \pi(440) t] $$ (b) Estimate all \(t\) -coordinates where \(P\) is maximum. (c) What does a person hear in addition to the frequencies of \(220,330,\) and \(440 \mathrm{Hz} ?\) (d) Graph the pressure produced by a speaker that can vibrate at \(110 \mathrm{Hz}\) and above.

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2-\sin 2 \theta=4 \sin 2 \theta$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$1-\sin x=\cos 2 x$$

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$4 \cos 2 \theta=8 \sin \theta \cos \theta$$
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