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Chapter 7: Problem 60

Write the product as a sum or difference. \(2 \cos 2 \theta \cos 5 \theta\)

Short Answer

Expert verified
The product can be written as a sum: \(\cos(3 \theta) + \cos(7 \theta)\)

Step by step solution

01

Apply product-to-sum identity

Apply the cosine product-to-sum identity to \(2 \cos 2 \theta \cos 5 \theta\), such that \(\alpha = 2 \theta\) and \(\beta = 5 \theta\). This means rewriting the expression as \(2 * \frac{1}{2}[\cos(2 \theta -5 \theta) + \cos(2 \theta + 5 \theta)]\)
02

Simplify the expression

Simplify the expressions within the brackets to get \(\cos(-3 \theta) + \cos(7 \theta)\)
03

Accounting for cosine's even property

Since cosine is an even function, meaning \(\cos(-x) = \cos(x)\), the expression \(\cos(-3 \theta)\) can be rewritten as \(\cos(3 \theta)\).

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Key Concepts

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

Product-to-Sum Identities
Trigonometric identities simplify complex expressions into more manageable forms. One important identity is the product-to-sum identity. This identity transforms a product of trigonometric functions into a sum or difference of trigonometric terms.
In this case, the product-to-sum identity for cosine is:
  • \(\cos \alpha \cos \beta = \frac{1}{2} [\cos (\alpha - \beta) + \cos (\alpha + \beta)]\)
Application: If you have an expression like \(2 \cos 2\theta \cos 5\theta\), you can first factor out the 2 and then use the product-to-sum identity to rewrite it. This results in:
  • \(2 * \frac{1}{2} [\cos (2\theta - 5\theta) + \cos (2\theta + 5\theta)]\)
This identity is incredibly useful for integrals and simplifying algebraic expressions involving trigonometric functions.
Cosine Function
The cosine function is a fundamental part of trigonometry. It relates the angle of a right triangle to the lengths of the adjacent side and the hypotenuse.

Key Properties of Cosine:
  • Range: The values of the cosine function range from -1 to 1.
  • Periodicity: Cosine has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
  • Symmetry: It is an even function, which will be discussed further below.
The cosine function forms the backbone of various trigonometric identities, allowing complex expressions to be manipulated into simpler forms for analysis and calculus.
Even Function Property
An even function is symmetric around the y-axis. This means it produces the same result for both positive and negative versions of the same input.

Cosine's Even Property:
  • For cosine, this property is expressed as: \(\cos(-x) = \cos(x)\).
In our problem, one of the expressions was \(\cos(-3\theta)\). Because cosine is even, we can rewrite this as \(\cos(3\theta)\).

Importance: This property simplifies problem-solving and allows mathematicians to manage trigonometric expressions more efficiently. Understanding and applying these symmetric properties help in reducing computation and enhancing clarity.

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Most popular questions from this chapter

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=6$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{6}$$

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle-5,-1\rangle\\\ &\mathbf{v}=\langle-1,1\rangle \end{aligned}$$

The vector \(\mathbf{u}=\langle 3240,2450\rangle\) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast food stand in one week. The vector \(\mathbf{v}=\langle 3.25,3.50\rangle\) gives the prices in dollars of the food items. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase prices by \(2 \frac{1}{2}\) percent.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)^{10}$$
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