Problem 20 - A Graphical Approach to Precalculus with Limits

Question

For e expression in Column I, choose the expression from Column II that completes an identity. You may have to rewrite one or both expressions. Do not use a calculator. \(\mathbf{I}\) \(\frac{\sec x}{\csc x}=\) _______ \(\mathbf{II}\) A. \(\frac{\sin ^{2} x}{\cos ^{2} x}\) B. \(\frac{1}{\sec ^{2} x}\) C. \(\sin (-x)\) D. \(\csc ^{2} x-\cot ^{2} x+\sin ^{2} x\) E. \(\tan x\)

Step-by-Step Solution

Answer

The identity completes with Option E: \(\tan x\).

Step 1: Rewrite the given expression using trigonometric identities

We start with the expression \(\frac{\sec x}{\csc x}\) from Column I. Remember that \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\). Rewriting using these identities, we have:\[\frac{\sec x}{\csc x} = \frac{\frac{1}{\cos x}}{\frac{1}{\sin x}} = \frac{1}{\cos x} \times \sin x = \frac{\sin x}{\cos x}\]

Step 2: Simplify the expression

The expression \(\frac{\sin x}{\cos x}\) simplifies to \(\tan x\) because the tangent of an angle is defined as the ratio of the sine to the cosine of that angle.

Step 3: Match the simplified expression with options from Column II

Now that we have simplified \(\frac{\sec x}{\csc x}\) to \(\tan x\), we look for an equivalent expression in Column II. Option E is \(\tan x\), which matches our simplified expression exactly.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental to understanding relationships within triangles, especially right-angled triangles. They offer a bridge between angles and ratios derived from triangle sides. The main trigonometric functions are:

• Sine (\( \sin \)): Ratio of opposite side to hypotenuse.
• Cosine (\( \cos \)): Ratio of adjacent side to hypotenuse.
• Tangent (\( \tan \)): Ratio of opposite to adjacent side, or \( \frac{\sin x}{\cos x} \).
• Cosecant (\( \csc \)): Reciprocal of sine, \( \frac{1}{\sin x} \).
• Secant (\( \sec \)): Reciprocal of cosine, \( \frac{1}{\cos x} \).
• Cotangent (\( \cot \)): Reciprocal of tangent, \( \frac{1}{\tan x} \).

Understanding these functions and their relationships helps in solving trigonometric identities and equations. Trigonometric identities, such as Pythagorean identities, offer equality statements involving the trigonometric functions, simplifying complex problems. Additionally, these identities play a vital role in calculus, physics, and engineering applications.

Expression Simplification

Expression simplification in mathematics involves rewriting an expression in a more comprehensible form. When dealing with trigonometric expressions, we often use identities to achieve simplification. Here is a step-by-step approach as illustrated in our problem:
1. **Identify known identities**: Recognize standard trigonometric identities like \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \).
2. **Substitute and rewrite**: Replace the trigonometric functions in the expression with their identity counterparts. In this problem, we start with \( \frac{\sec x}{\csc x} \), rewriting it using identities, leading to \( \frac{\frac{1}{\cos x}}{\frac{1}{\sin x}} \).
3. **Simplify fractions**: Simplify the resultant fraction by multiplying or canceling terms when possible. Here, it becomes \( \frac{1}{\cos x} \times \sin x = \frac{\sin x}{\cos x} \).
4. **Match simplified expression**: Convert the simplified fraction to its equivalent trigonometric function such as \( \tan x \). In this way, simplification helps equate our expression to one of the given choices.

Sine and Cosine Ratios

Sine and cosine ratios form the foundation for understanding other trigonometric functions and identities. These ratios are not only crucial for solving triangles but also in mapping algebraic identities into simpler forms. Here’s how they are applied in our exercise:

• **Sine (\( \sin x \))**: Represents the y-coordinate in the unit circle context, important in converting expressions through Pythagorean identities.
• **Cosine (\( \cos x \))**: Represents the x-coordinate in the unit circle, helps relate other trigonometric functions via angles.
• **Tangent (\( \tan x \)) as a Ratio**: Defined as \( \frac{\sin x}{\cos x} \), showcasing how the sine and cosine interplay. This relationship is useful in converting the expression \( \frac{\sin x}{\cos x} \) directly into \( \tan x \).

Whenever sine and cosine are involved, we tap into their complementary behavior, especially evident in the identity \( \sin^2 x + \cos^2 x = 1 \). Understanding these core definitions allows for not only solving identities but also working with complex trigonometric expressions intuitively.