Problem 139 - Precalculus with Limits

Question

In Exercises 139 and 140, determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \( u, \sin 2u = -2 \sin u \cos u \).

Step-by-Step Solution

Answer

The statement provided is false because it contradicts the identity \( \sin 2u = 2 \sin u \cos u \) and doesn't adhere to the properties of odd functions.

Step 1: Identify the Double Angle Identity for Sine

The double angle identity for the sine function is given by \( \sin 2u = 2 \sin u \cos u \). This is a standard trigonometric identity.

Step 2: Understand the properties of odd functions

Odd functions follow a specific property, that is \( f(-x) = -f(x) \). You need to consider this while observing the statement.

Step 3: Apply the property of odd functions to \( \sin u \)

When you apply the property of odd functions to \( \sin u \), you get \( \sin(-u) = -\sin(u) \). Now, you can test the given statement by substituting \( u \) with \( -u \).

Step 4: Test the statement by substitution

The given statement is \( \sin 2u = -2 \sin u \cos u \). Substituting \( u \) with \( -u \), you get \( \sin -2u = -2 \sin -u \cos -u \). By applying the properties of odd functions, this simplifies to \( -\sin 2u = 2 \sin u \cos u \)

Step 5: Compare the original identity and the derived expression

The original double angle identity is \( \sin 2u = 2 \sin u \cos u \). Comparing it with the expression derived from the given statement, \( -\sin 2u = 2 \sin u \cos u \), you can see that they are not the same. The given statement is therefore false.

Key Concepts

Odd Functions

Understanding the concept of odd functions is crucial when dealing with trigonometric identities. An odd function is defined by a particular symmetrical behavior. Specifically, an odd function satisfies the condition

• \( f(-x) = -f(x) \)

This means if you plug in a negative value for the variable, the output will be the negative of the output for the positive value.
The sine function, \( \sin(x) \), is a classic example of an odd function. If you were to calculate \( \sin(-u) \), you would get \(-\sin(u)\).
This property of sine plays a critical role in simplifying expressions and solving trigonometric equations. Understanding this symmetry helps identify incorrect propositions, such as the claim that \( \sin(2u) = -2 \sin(u) \cos(u) \) when \( u \) is negative.

Double Angle Identity

The double angle identities are powerful tools in trigonometry that simplify expressions involving angles twice as large as a given angle. The double angle identity for sine is given by

• \( \sin(2u) = 2 \sin(u) \cos(u) \)

This identity allows you to expand the sine of a double angle into a product of sine and cosine of the original angle.
Using this identity, you can simplify complex trigonometric expressions and solve equations more efficiently.
In verifying or debunking trigonometric statements, it’s important to compare these identities directly with the claims. If the double angle identity \( \sin(2u) = 2 \sin(u) \cos(u) \) doesn't match a statement, then the statement might be incorrect, like in this exercise.

Sine Function Properties

The sine function exhibits several important properties that are essential for solving trigonometric problems. Among these properties, the odd function characteristic is quite vital. The sine function satisfies

• \( \sin(-u) = -\sin(u) \)

This indicates the sine’s odd symmetry and helps when dealing with negative angles.
Another critical property is related to its periodic behavior. Sine has a period of \(2\pi\), meaning its values repeat every \(2\pi\) units.
Understanding these properties is crucial when evaluating the correctness of trigonometric identities and statements.
For instance, given a statement involving \( \sin(2u) \) and negative inputs, knowing these properties allows you to test and determine the validity logically.