Problem 80 - Algebra and Trigonometry

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

After graphing the equations, it can be observed that the graphs don't coincide indicating that this equation is not an identity. After setting \(\sin x=-\cos x \tan (-x)\), x value(s) can be found where both sides of the equation are defined but not equal.

Step 1: Graphing the Equations

Start by graphing both sides of the equation \(\sin x\) and \(-\cos x \tan(-x)\) on the same graph. This can be done using graphing calculator software or plotting methods in mathematics. You will observe two waves generated from these graphs.

Step 2: Analyzing the Graphs

After graphing each side of the equation, observe whether these two graphs overlap or not. If they coincide fully at every point, that indicates the equation is an identity because the outputs of both sides of the equation are the same for every input value of x. However, if they do not coincide, that means the equation is not an identity because there are values of x for which the outputs of both sides of the equation are not the same.

Step 3: Verification

If the graphs coincide, verify that the equation is an identity by comparing the two equations. For instance, using trigonometric identities or properties to transform one side of the equation into the other. If it's possible to make the transformation, then it's an identity.

Step 4: Find x value

Finally, if the graphs of \(\sin x\) and \(-\cos x \tan(-x)\) do not coincide, find where both sides are defined but not equal. A value of x is needed where both \(\sin x\) and \(-\cos x \tan(-x)\) give different results. Solve for x by setting \(\sin x=-\cos x \tan (-x)\) and find the x value that gives different results for each side of the equation

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions is an essential skill in understanding the behavior of sine, cosine, and other trigonometric functions. To begin, you need a graphing calculator or graphing software which allows you to plot functions in the same coordinate plane. This exercise focuses on visualizing the functions \( \sin x \) and \( -\cos x \tan (-x) \).

When graphing these, you will see their characteristic wave patterns. Sine and cosine functions have regular wave-like patterns that repeat every \( 2\pi \).

• For \( \sin x \), the wave peaks at 1 and troughs at -1.
• For \( \cos x \), it also peaks at 1 and troughs at -1 but starts at 1 when \( x = 0 \).

Understanding these basic patterns helps in identifying whether two trigonometric expressions are equal over a range of x values. This visual tool aids in recognizing equations that are identities across their entire domain.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and are solved over specific intervals. In this exercise, the equation \( \sin x = -\cos x \tan(-x) \) requires examination to determine if it holds for all values of \( x \), or just specific ones.

To solve trigonometric equations, it's useful to apply identities and transformations:

• Start by evaluating both sides of the equation. Notice the structure of \( \tan(-x) \) which equals \(-\tan(x)\) by the odd function property of tangent.
• Swap known identities or use algebraic manipulation to try to simplify or equate the functions.

The goal is to determine if there's a broad set of x values that satisfy the equation, indicating it's an identity, or just a few where this condition holds.

Verification of Identities

Verifying trigonometric identities requires testing if two expressions are equal for all values of the variable involved. In our exercise, verifying \( \sin x = -\cos x \tan (-x) \) involves more than graph observations.

Here's what to do:

• Start by simplifying the complex side \(-\cos x \tan(-x)\) using known trigonometric identities.
• Recognize that if algebraic manipulation or identities transform one side into the other, it confirms they're identical.

It's crucial to remember that if an equation does not simplify into a known identity, there may still be values of \( x \) where both sides give the same result without being an identity.

To firmly conclude if the equation is indeed an identity, you may have to check specific values where the equation holds true or disproves it by finding values where it diverges. Ultimately, this task sharpens your algebra skills and enhances understanding in analysing trigonometric behavior.