Problem 37 - Contemporary Precalculus

Question

Write each expression as a sum or difference. $$\sin 17 x \sin (-3 x)$$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$\sin 17x \sin (-3x)$$ as a sum or difference. Answer: $$-\sin(17x) \cos (3x) + \cos(17x) \sin (3x)$$

Step 1: Identify the angle arguments and determine the identity to use

We have the expression, $$\sin 17x \sin (-3x)$$, which we need to rewrite as a sum or difference. Notice that $$\sin (-3x) = -\sin(3x)$$ as sine function values for negative angles are negative. So, the given expression can be written as $$\sin 17x (-\sin 3x)$$, which indicates that we need to use the second sine identity (angle difference identity).

Step 2: Rewrite the expression using the sine angle difference identity

Let's rewrite the expression using the sine angle difference identity: \(\sin (A - B) = \sin A \cos B - \cos A \sin B\) Here, $$A = 17x$$ and $$B = 3x$$. Therefore, $$\sin(17x - 3x) = \sin(17x) \cos (3x) - \cos(17x) \sin (3x)$$

Step 3: Write the given expression as a sum or difference

Now we can rewrite the given expression as a sum or difference using the identity we found: $$\sin 17x \sin (-3x) = -\sin 17x \sin 3x = -[\sin(17x) \cos (3x) - \cos(17x) \sin (3x)]$$ The final expression as a sum or difference is: $$-\sin(17x) \cos (3x) + \cos(17x) \sin (3x)$$

Key Concepts

Angle Sum and Difference Formulas

The Angle Sum and Difference Formulas are essential tools in trigonometry. These formulas help us expand the sine, cosine, and tangent of sums or differences of angles into simpler trigonometric expressions. When dealing with complex expressions, they allow us to break them down into more manageable components.
For the sine function, the angle difference identity is given by:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

This identity transforms a difference of angles, \(A - B\), into a combination of sine and cosine functions.

The formula for the cosine of angle differences is:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

These identities are essential when you need to rewrite expressions involving sums or differences into a simpler format. It often simplifies computations and can make solving equations more straightforward.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They measure the vertical and horizontal distances from the origin for angles on the unit circle.
For any angle \( \theta \), the sine (\( \sin \theta \)) is the y-coordinate, while the cosine (\( \cos \theta \)) is the x-coordinate of the point on the unit circle corresponding to \( \theta \).

These functions are periodic, meaning they repeat their values in a regular pattern. The sine function has a period of \(2\pi\), as does the cosine function. This periodicity is key to predicting the values of these functions for various angles:

• Sine Function: \( \sin \theta = \text{opposite} / \text{hypotenuse} \)
• Cosine Function: \( \cos \theta = \text{adjacent} / \text{hypotenuse} \)

Understanding these basic definitions provides the backbone for using the Angle Sum and Difference Formulas effectively.

Negative Angle Identity

The Negative Angle Identity is a property of trigonometric functions that describes how to handle negative angles. Knowing this helps you work with expressions involving negative angles across different trigonometric functions.
For the sine and cosine functions, the identities are:

• \( \sin(-\theta) = -\sin(\theta) \)
• \( \cos(-\theta) = \cos(\theta) \)

The sine function is odd, meaning it flips its sign when the angle is negated. Conversely, the cosine function is even, keeping its value regardless of the negation.

This means when you see the sine of a negative angle, you simply take the sine of the positive angle and change its sign. For a cosine of a negative angle, nothing changes. Understanding these rules simplifies working with Angle Sum and Difference Formulas, especially when negative angles are involved.