Problem 5 - Contemporary Precalculus

Question

Insert one of \(A-F\) on the right of the equal sign so that the resulting equation appears to be an identity when you test it graphically. You need not prove the identity. A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\) \(\csc x \tan x=\) ____________

Step-by-Step Solution

Answer

__ Answer: B. \(\sec x\)

Step 1: Identify algebraic expressions for functions

First, let's remember the definitions for the trigonometric functions: \(\csc x = \frac{1}{\sin x}\) and \(\tan x = \frac{\sin x}{\cos x}\)

Step 2: Simplify the equation

Now, let's simplify the left side of the equation by substituting the definitions of \(\csc x\) and \(\tan x\): \(\csc x \tan x = \frac{1}{\sin x} \times \frac{\sin x}{\cos x} = \frac{1}{\cos x}.\) Notice that the left side simplifies to \(\frac{1}{\cos x}\). We can now compare it with the given options: A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\)

Step 3: Identifying the best match

From the list, we can identify that option B, \(\sec x\), is the reciprocal of \(\cos x\), which means: \(\sec x = \frac{1}{\cos x}\) Now we have found the expected match: \(\csc x \tan x =\sec x\)

Step 4: Testing the answer graphically

For the verification purposes, you can plot both functions (\(\csc x \tan x\) and \(\sec x\)) graphically side by side, and observe whether they appear to be identical. You can use an online graphing calculator or your calculator for this. If they appear to be identical, then it confirms that the answer is indeed correct.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived by taking the reciprocal or the inverse of basic trigonometric functions such as sine, cosine, and tangent. These functions are essential tools in trigonometry.
Here is a breakdown:

• **Cosecant ( \( \csc x \) )**: This is the reciprocal of the sine function. Mathematically, it’s defined as \( \csc x = \frac{1}{\sin x} \).
• **Secant ( \( \sec x \) )**: This is the reciprocal of the cosine function. It’s represented as \( \sec x = \frac{1}{\cos x} \).

Understanding these reciprocal relationships helps with the simplification of trigonometric expressions and proving identities.
When you encounter a problem involving these functions, begin by expressing them in terms of sine and cosine. This simplification can make it easier to work through complex expressions.

Graphical Verification

Graphical verification is a technique used to confirm that two trigonometric expressions are identical. It's a quick way to check if an identity holds true.
Here is how you can do this:

• **Plot both expressions**: Use a graphing calculator or software to plot both sides of your trigonometric equality on the same set of axes.
• **Compare the graphs**: Check whether the plotted graphs appear to be identical over a reasonable domain. They should overlay completely if the expressions are identical.

If the graphs are indistinguishable from each other, it provides a strong visual confirmation that the identity is correct.
Graphical methods are especially useful in educational settings where visual confirmation can aid in understanding complex equations. It should be noted, though, that graphical verification is not rigorous proof, but a very useful check.

Trigonometric Simplification

Simplifying trigonometric expressions involves reducing them to their most basic form using known identities and algebraic techniques.
Key steps in simplification include:

• **Substituting Known Identities**: Use identities like \( \csc x = \frac{1}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \) to replace complex expressions with simpler ones.
• **Canceling Terms**: After substituting, see if there are terms in both the numerator and the denominator that can cancel out. This simplifies the entire expression substantially.

By following these steps, an expression like \( \csc x \tan x \) simplifies to \( \frac{1}{\cos x} \), showing the simplicity behind what initially seems complex.
Simplified expressions are not only easier to work with in calculations, but are also crucial for solving problems in calculus and other advanced mathematical studies.