Problem 80 - 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. $$\sin x=-\cos x \tan (-x)$$

Step-by-Step Solution

Answer

Whether \(\sin x = -\cos x \tan(-x)\) is an identity can be evaluated by graphing both sides. If not an identity, a certain value of 'x' where both sides of the equation are defined but unequal needs to be found.

Step 1: Analyze the equation

First, rewrite the equation in a clearer form: \(\sin x = -\cos x \cdot \tan(-x)\). Take a look at the right hand side of equation: it seems that it has a product of cosine and tangent functions, but the signs of 'x' in each function are different, which may affect the equality of both sides. Before graphing, pay attention to this detail.

Step 2: Graphing the equation

Graph each side of \(\sin x = -\cos x \cdot \tan(-x)\) separately in the same viewing rectangle. Observe carefully whether the two sides coincide.

Step 3: Analyze the graphs

If the two graphs coincide all along the 'x' axis, the equation is an identity. Otherwise, it isn't. Make sure to inspect carefully as there might be some values of 'x' for which both graphs do match but others for which they don't.

Step 4: Verify the identity

In the case of the graphs not fully coinciding, find a value of \(x\) for which both sides are defined but not equal. This will be a solid proof that the given equation is not an identity. Since both functions are periodic, the value of 'x' might lay outside of the standard range, [-π/2, π/2].

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can help visualize relationships or verify equations. To graph functions like \( \sin x \) and \( -\cos x \cdot \tan(-x) \), it's essential to understand the behavior of each component.

• Sine Function: \( \sin x \) has a wave-like pattern, repeating every \( 2\pi \). It starts at 0, peaks at \( \pi/2 \), hits 0 at \( \pi \), and reaches -1 at \( 3\pi/2 \).
• Cosine Function: \( \cos x \) shares a similar wave pattern, but it starts at 1 and decreases to -1 over the same interval.
• Tangent Function: \( \tan(-x) \) is intriguing because of its negative argument, causing reflection over the y-axis. It has asymptotes at every odd multiple of \( \pi/2 \), showing steep increases and decreases.

When graphing, plot each function individually and note how they overlap. Matching graphs indicate potential identities, but further verification is necessary. Always use a consistent scale and a range that captures periodic behavior.

Product of Functions

The product of trigonometric functions often requires careful manipulation and understanding. In this exercise, \( -\cos x \cdot \tan(-x) \) is the product of two distinct functions, each with its own properties.

• Negative Cosine: Multiplying \( \cos x \) by -1 reflects its graph over the x-axis. This change needs to be considered when figuring out how it interacts with tangent.
• Product with Tangent: \( \tan(-x) = -\tan x \), meaning the product actually simplifies to \( \cos x \cdot \tan x \).

Interpreting this, the product simplifies the equation and effects can be analyzed graphically. When two functions interact, it can create complex patterns—sometimes complicating equations more than simplifying. Understanding each function’s role is crucial in these scenarios.

Trigonometric Equations

Solving trigonometric equations like \( \sin x = -\cos x \cdot \tan(-x) \) involves identifying solutions where both sides are equal. To assess if this is true for all \( x \), follow these steps:

• Check Known Identities: Look for known identities that can simplify the equation. Often, simplifications reveal whether one side can equal the other.
• Graphical Analysis: As described earlier, graph each side to compare visually. If the graphs coincide perfectly, it may be an identity.
• Identify Counterexamples: If not an identity, find a specific value of \( x \) where the equality breaks. This proves the equation fails for some values.

Given periodic behavior, verify across multiple cycles. For example, checking within one period may not suffice; diversity in testing strengthens conclusions. Testing different methods—algebraic manipulations, graphical checks, or known identities—ensures comprehensive understanding.