Problem 68 - Precalculus

Question

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. $$ \sin (\pi-x) $$

Step-by-Step Solution

Answer

\( \sin(\pi - x) = \sin x \). Graphs are identical.

Step 1: Understanding the Trigonometric Identity

To simplify the expression \( \sin(\pi - x) \), we use the trigonometric identity for sine: \( \sin(\pi - x) = \sin \pi \cos x - \cos \pi \sin x \). Knowing that \( \sin \pi = 0 \) and \( \cos \pi = -1 \), this simplifies to: \( 0 \, \cdot \, \cos x - (-1) \, \cdot \, \sin x \).

Step 2: Simplifying the Expression

After applying the identity and known values, the expression \( \sin(\pi - x) \) simplifies to \( \sin x \). This step confirms that the expression can be simplified using trigonometric identities.

Step 3: Graphing and Verifying

Now, graph both \( \sin(\pi - x) \) and \( \sin x \) on the same axes. Upon graphing, you should observe that both functions produce the same sine wave. This confirms the correctness of the identity and simplification.

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is a periodic function, which means it repeats its pattern at regular intervals. The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse. In mathematical expressions, it is denoted as \( \sin \theta \). Here's what you need to remember about sine functions:

• The range of the sine function is between -1 and 1.
• Sine is an odd function, which means \( \sin(-x) = -\sin(x) \).
• It has a period of \( 2\pi \), which means \( \sin(x) = \sin(x + 2\pi) \).

Understanding these properties can help in simplifying expressions involving sine, as we can see in \( \sin(\pi-x) \). By recognizing identities and properties, such as those involving the reference angle \( \pi - x \), we can transform and simplify these expressions effectively.

Graphing Functions

Graphing functions is a visual way to understand how different trigonometric expressions compare or simplify to each other. When graphing the sine function or any trigonometric function, consider the following tips:

• Identify the amplitude, which affects the height of the wave (for sine, this is usually 1).
• Determine the period, which affects the frequency of repetition (for sine, it's \( 2\pi \)).
• Look for phase shifts, which can move the graph left or right, depending on the expression.
• Note vertical shifts, which can move the graph up or down.

To verify identities like \( \sin(\pi - x) = \sin x \), plot both expressions on the same coordinate system. You'll see that they produce identical curves, showing that they are indeed the same function. This visual confirmation is a powerful tool to validate mathematical identities and simplifications.

Expression Simplification

Simplifying expressions in trigonometry often requires knowledge of trigonometric identities. A trigonometric identity is an equation that is true for all values of the variable where both sides of the equation are defined. In the case of \( \sin(\pi - x) \), we used the identity:

• \( \sin(\pi - x) = \sin \pi \cos x - \cos \pi \sin x \)

By substituting known values \( \sin \pi = 0 \) and \( \cos \pi = -1 \), the expression simplifies to \( \sin x \). Here is the process broken down:

• Apply the identity: \( \sin(\pi - x) = \sin \pi \cos x - \cos \pi \sin x \)
• Insert known values: becomes \( 0 \cdot \cos x + 1 \cdot \sin x \)
• Simplified result: Simply \( \sin x \)

These steps highlight how trigonometric identities provide a means to reduce complex expressions to simpler ones, thereby making calculations easier and enabling further mathematical analysis.