Problem 35 - Algebra 2

Question

You can determine whether or not an equation may be a trigonometric identity by graphing the expressions on either side of the equals sign as two separate functions. If the graphs do not match, then the equation is not an identity. If the two graphs do coincide, the equation might be an identity. The equation has to be verified algebraically to ensure that it is an identity. $$ \frac{1}{\sec x}+\frac{1}{\csc x}=1 $$

Step-by-Step Solution

Answer

The equation is not a trigonometric identity as it doesn't hold for all x.

Step 1: Rewrite in Terms of Sine and Cosine

First, we'll express the trigonometric functions in terms of sine and cosine. We know that \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). This means that \( \frac{1}{\sec x} = \cos x \) and \( \frac{1}{\csc x} = \sin x \). Substitute these into the equation: \( \cos x + \sin x = 1 \).

Step 2: Graph the Functions

Graph the left-hand side function \( f(x) = \cos x + \sin x \) and the right-hand side constant \( g(x) = 1 \) to compare them visually. Observe whether the two graphs overlap completely for all values in their domain.

Step 3: Analyze Graph Results

Upon graphing, you will notice that \( f(x) = \cos x + \sin x \) and \( g(x) = 1 \) do not coincide everywhere; they overlap only at specific points. This means the given equation doesn't hold for all values of \( x \).

Step 4: Verifying Algebraically

Since the equation is not consistent across all values, let's ensure by algebraically checking: \( \cos x + \sin x = 1 \). Try a simple value like \( x = 0 \): \( \cos 0 + \sin 0 = 1 + 0 = 1 \), which holds, but try \( x = \frac{\pi}{2} \): \( \cos \frac{\pi}{2} + \sin \frac{\pi}{2} = 0 + 1 = 1 \). Hence, the equation is true for specific angles but not generally an identity.

Key Concepts

Graphing Functions

Graphing functions is a powerful visual tool for understanding equations and their solutions. In context here, it means you plot the expressions on each side of the equation as separate functions on a graph. Visually comparing these graphs can provide insights into whether the equation might be a trigonometric identity. If the graphs are identical across their entire domain, the equation is a candidate for being an identity. Steps to Graph Functions for Verification:

• Identify the functions on each side of the equation. Rewrite any complex expressions in simpler forms if necessary.
• Use graphing software or plot on graph paper to visualize both functions.
• Carefully observe whether the functions coincide exactly for all values of the variable or only at certain points.

If the graphs do not overlap everywhere, it indicates that the equation is not a trigonometric identity throughout the domain. This shows the importance of graphing in quickly highlighting potential issues without performing detailed algebraic proofs first.

Sine and Cosine Functions

Understanding sine and cosine functions is crucial for rewriting complex trigonometric expressions. These functions are periodic and represent the fundamental ratios in trigonometry.Quick Review of Sine and Cosine:

• Sine function, denoted as \( \sin x \), gives the y-coordinate of the point on the unit circle corresponding to an angle x.
• Cosine function, written as \( \cos x \), provides the x-coordinate of such a point.
• Both functions oscillate between -1 and 1.
• Their periodic nature is characterized by their repetition over intervals of \( 2\pi \).

In the provided exercise, you convert other trigonometric terms like secant (\( \sec x = \frac{1}{\cos x} \)) and cosecant (\( \csc x = \frac{1}{\sin x} \)) into sine and cosine. This simplification helps in analyzing the equation more effectively and can make it easier to identify potential identities.

Algebraic Verification

Algebraic verification is a method used to confirm the true identity of a trigonometric equation. It involves manipulating the equation to see if you can prove it holds for all values in the domain, unlike graphing, which might only suggest possibilities due to visual overlap at specific points.Steps to Verify Algebraically:

• Start by simplifying each side of the equation using basic algebraic transformations. Consider known trigonometric identities for substitution.
• Check the equation by substituting common angles like 0, \(\pi/2\), \(\pi\), etc., and see if the equation holds true.
• Use mathematical logic to test whether the simplifications yield an identity like \( 0=0 \) or any other universally true statement.

In the exercise, algebraic verification with common values such as \( x = 0 \) and \( x = \pi/2 \), showed the equation held true for specific angles but not all, confirming it was not a universal identity. This method acts as a thorough check beyond the quick results provided by graphical methods.