Problem 81 - Precalculus Essentials

Question

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin \left(x+\frac{\pi}{4}\right)=\sin x+\sin \frac{\pi}{4}$$

Step-by-Step Solution

Answer

The equation \( \sin \left(x+\frac{\pi}{4}\right) = \sin x + \sin \frac{\pi}{4} \) is not an identity as the graphs of both sides do not coincide. A value of x=0 makes both sides defined but not equal.

Step 1: Understanding Trigonometric Identities

A trigonometric identity is an equation involving trigonometric ratios of an angle, which is true for all values of the variables. In this case our equation is \( \sin \left(x+\frac{\pi}{4}\right) = \sin x + \sin \frac{\pi}{4} \).

Step 2: Plot both sides of the equation

We need to graph both sides of the given equation: \(s1 = \sin (x + \frac{\pi}{4})\), and \(s2 = \sin x + \sin \frac{\pi}{4}\). Analyzing them we can determine if the graphs coincide to prove the equation an identity or if they are different, disproving the equation as an identity.

Step 3: Analyze the graphs

Upon graphing the two sides, the graphs do not coincide at all x-values which indicates that the equation is not an identity.

Step 4: Determine a point of difference between both sides

Choose a value of x for which the two sides of the equation are both defined but are not equal. Here, x=0 is a good choice, because \( \sin \left(0+\frac{\pi}{4}\right) = \sin \frac{\pi}{4} \neq \sin 0 + \sin \frac{\pi}{4} \). So for x=0, the equation doesn't hold.

Key Concepts

Sin (x + pi/4) Identity

Understanding the sine of a sum, such as when we encounter the expression \( \sin (x + \frac{\pi}{4}) \), is crucial for solving trigonometric problems. This expression is based on the sine sum identity, which states:
\[ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]
Applying this identity to our given problem, we have:
\[ \sin \left(x + \frac{\pi}{4}\right) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} \]
Since \( \cos \frac{\pi}{4} \), also known as \( \sin \frac{\pi}{4} \), equals \( \frac{\sqrt{2}}{2} \), the equation simplifies to:
\[ \sin \left(x + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\sin x + \frac{\sqrt{2}}{2}\cos x \]
This shows the error in the original equation \( \sin (x + \frac{\pi}{4}) eq \sin x + \sin \frac{\pi}{4} \), because the right side does not consider the product of sine and cosine that comes from the sine sum identity.

Verifying Trigonometric Equations

To verify a trigonometric equation means to prove that the equation is valid for all permissible values of the variable(s) involved. Verifying is an essential skill, as it allows you to distinguish between identities and non-identities. Here's how verification commonly unfolds:

• Apply known trigonometric identities to simplify the equation.
• Analyze if the simplified expression holds true for all values within the variable's domain.
• If any step fails to prove the identity for all values, then the equation is not an identity.

To verify the original problem's equation, an attempt to simplify it using trigonometric identities should be made. If simplification leads to an equivalent expression on both sides of the equation for all values of \( x \), then the equation is an identity. In our case, once simplified correctly, it's clear that \( \sin (x + \frac{\pi}{4}) \), does not simplify to \( \sin x + \sin \frac{\pi}{4} \). Therefore, we can conclude that the original equation is not an identity.

Graphing Trigonometric Functions

Graphing is a potent tool for understanding and verifying trigonometric equations. By graphing both sides of an equation separately and comparing the results, we can visually assess whether an equation is an identity. Here are steps to graph functions effectively:

• Ensure a proper scale and viewing window to capture the behavior of the functions.
• Plot each function with careful consideration for key points, like intercepts and maximums/minimums.
• Look for overlaps or distinctions between the graphs.

In the context of our problem, graphing \( \sin (x + \frac{\pi}{4}) \), and \( \sin x + \sin \frac{\pi}{4} \), will clearly show that they do not overlap completely. This visual discrepancy further confirms that the given equation is not an identity. The graph illustrates that the functions disagree, notably where \( x = 0 \), which we highlighted during the verification process. Thus, graphing can be a definitive method for disproving false trigonometric identities.