Problem 6 - 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. \(\frac{1}{\sin x \cos x}\) \(\frac{\sin x}{\tan x}=\) _____________

Step-by-Step Solution

Answer

Answer: A. \(\cos x\)

Step 1: Simplify the given expression

To simplify the given expression \(\frac{\sin x}{\tan x}\), we will first need to write \(\tan x\) in terms of \(\sin x\) and \(\cos x\). Recall that \(\tan x = \frac{\sin x}{\cos x}\), so, we can replace \(\tan x\) in the given equation: \(\frac{\sin x}{\tan x} = \frac{\sin x}{\frac{\sin x}{\cos x}}\)

Step 2: Perform the division

Now we will perform the division in the expression. Dividing by a fraction is the same as multiplying by its reciprocal, so: \(\frac{\sin x}{\frac{\sin x}{\cos x}} = \sin x \times \frac{\cos x}{\sin x}\)

Step 3: Simplify the expression

From the previous step, we observe that \(\sin x\) in the numerator and denominator cancel each other out, leaving us with just the remaining term: \(\sin x \times \frac{\cos x}{\sin x} = \cos x\)

Step 4: Compare the simplified expression with the given options

Now that we have a simplified expression \(\cos x\), we will look through the given options and choose the one that makes the equation an identity: A. \(\cos x\) B. \(\sec x\) C. \(\sin ^{2} x\) D. \(\sec ^{2} x\) E. \(\sin x-\cos x\) F. \(\frac{1}{\sin x \cos x}\) Comparing the simplified expression with all the given options, the correct choice that makes the equation an identity is Option A. Therefore, the answer is: \(\frac{\sin x}{\tan x}=\cos x\)

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics. They help us understand and represent relationships in right triangles and circles.
When we work with them, we often use a few key functions:

• Sine (\(\sin x\)): Measures the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\(\cos x\)): The ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan x\)): Calculated as the sine divided by cosine (\(\tan x = \frac{\sin x}{\cos x}\)).

This exercise focused on these basic trigonometric functions to simplify a given expression. Understanding the relationships between these functions is crucial for tackling various mathematical problems.

Identity Verification

Verifying identities in trigonometry is like solving a puzzle. You aim to show that two expressions are the same. In this exercise, we started with \(\frac{\sin x}{\tan x}\) and tried to show it equals one of the given options.
Verification steps involve:

• Simplifying one side to match the other.
• Using basic trigonometric identities like \(\tan x = \frac{\sin x}{\cos x}\).
• Cancelling common terms to simplify the expression further.

In our example, simplifying led us to match the left side with \(\cos x\), validating it as the correct identity from the options. This process requires a mix of algebraic manipulation and understanding of trigonometric properties.

Simplifying Expressions

Simplifying trigonometric expressions involves using known identities to reduce complexity.
Here’s a breakdown of how we simplified \(\frac{\sin x}{\tan x}\):

• Substitution: Replace \(\tan x\) with \(\frac{\sin x}{\cos x}\).
• Division by a Fraction: Dividing by a fraction is akin to multiplying by its reciprocal.
• Cancellation: The \(\sin x\) terms cancelled, simplifying the expression to \(\cos x\).

By breaking down the expression into simpler parts, we found the matching identity and confirmed it as \(\cos x\). This approach makes understanding the problem more manageable and highlights the importance of mastering simplification techniques.