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Chapter 7: Problem 170

Rewrite the product as a sum or difference. $$\sin (-x) \sin (5 x)$$

Short Answer

Expert verified
\(\frac{1}{2}[\cos(6x) - \cos(4x)]\)

Step by step solution

01

Formula for Product-to-Sum

We are given the product \(\sin(a) \sin(b)\). The product-to-sum formula for sine is: \(\sin(a)\sin(b) = \frac{1}{2}\left[\cos(a-b) - \cos(a+b)\right].\)
02

Identify Parameters

Identify terms \(a\) and \(b\) from the expression \(\sin(-x) \sin(5x).\)Here, \(a = -x\) and \(b = 5x\).
03

Apply the Formula

Substitute \(a = -x\) and \(b = 5x\) into the product-to-sum formula:\[\sin(-x)\sin(5x) = \frac{1}{2}\left[\cos((-x) - 5x) - \cos((-x) + 5x)\right].\]
04

Simplify the Expression

Calculate \((-x) - 5x = -6x\) and \((-x) + 5x = 4x\).Substitute these into the formula:\[\frac{1}{2}\left[\cos(-6x) - \cos(4x)\right].\]
05

Use Even Property of Cosine

Recall that cosine is an even function, meaning \(\cos(-\theta) = \cos(\theta).\)Therefore, \(\cos(-6x) = \cos(6x)\).
06

Final Expression

Substitute the even property result back:\[\frac{1}{2}\left[\cos(6x) - \cos(4x)\right].\]This is the expression for \(sin(-x) \sin(5x)\) rewritten as a difference.

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Key Concepts

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

Product-to-Sum Formula
To transform the product of two trigonometric functions into a sum or difference, we use the product-to-sum formula. This formula is especially useful in simplifying expressions involving sine and cosine functions. The product-to-sum formula for the product of sines is given by:
  • \(\sin(a)\sin(b) = \frac{1}{2}\left[\cos(a-b) - \cos(a+b)\right]\)
This formula allows us to break down the product of sine functions into a manageable form, which often makes solving equations or integration easier. In the original exercise, this formula was used to convert the expression \(\sin(-x)\sin(5x)\) into a difference of cosines.
Sine Function
The sine function is a crucial concept in trigonometry, representing the y-coordinate of a point on the unit circle as the angle cycles through 360 degrees or \(2\pi\) radians. Some important properties of the sine function include:
  • Sine is a periodic function with a period of \(2\pi\).
  • The function is odd, meaning it satisfies \(\sin(-x) = -\sin(x)\).
  • Its range is between -1 and 1, covering all possible sine values of angles.
In this exercise, \(\sin(-x)\) is observed as an odd function, and this property is essential for subsequent steps in simplifying the expression using the product-to-sum formula.
Cosine Function
The cosine function, like sine, plays a foundational role in trigonometry. It gives the x-coordinate of a point on the unit circle. Some critical attributes of the cosine function include:
  • It is an even function, satisfying the property \(\cos(-x) = \cos(x)\).
  • Cosine is also periodic with a period of \(2\pi\).
  • The range of cosine is also between -1 and 1, analogous to sine.
In this exercise, the cosine function's property of being even is utilized to simplify the expression from \(\cos(-6x)\) to \(\cos(6x)\), allowing for a more straightforward expression of the final result.
Even Function Property
An even function is symmetric about the y-axis, which means its graph does not change when reflected over this axis. The defining property of an even function is:
  • \(f(-x) = f(x)\)
The cosine function is a classic example of an even function. Applying this characteristic to our problem, the term \(\cos(-6x)\) simplifies to \(\cos(6x)\). This not only simplifies the expression itself but also the calculation process, as understanding which trigonometric identities are even or odd helps to predict behavior and simplify computations.

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Most popular questions from this chapter

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