Problem 13 - A Graphical Approach to Precalculus with Limits

Question

For expression in Column I, choose the expression from Column II that completes a fundamental identity. Do not use a calculator. \(\mathbf{I}\) \(\frac{\cos x}{\sin x}=\)_____ \(\mathbf{I}\) A. \(\sin ^{2} x+\cos ^{2} x\) B. cot \(x\) C. \(\sec ^{2} x\) D. \(\frac{\sin x}{\cos x}\) E. \(\cos x\)

Step-by-Step Solution

Answer

The expression is \( \cot x \) (option B).

Step 1: Identify the Given Expression

We are given the expression \( \frac{\cos x}{\sin x} \). Our goal is to recognize this as a trigonometric identity.

Step 2: Recall Trigonometric Identities

One of the basic trigonometric identities is that \( \frac{\cos x}{\sin x} \) is equal to \( \cot x \), which is the cotangent of \( x \). The cotangent is the reciprocal of the tangent function.

Step 3: Match to Column II

Looking at Column II, we find option B \( \cot x \) matches our expression from Step 2, \( \frac{\cos x}{\sin x} = \cot x \).

Step 4: Verify the Match

Since \( \cot x = \frac{\cos x}{\sin x} \), this confirms that the expression completes the fundamental identity of the cotangent function.

Key Concepts

Cotangent

Cotangent is one of the six fundamental trigonometric functions. It is denoted as \( \cot x \), where \( x \) is an angle in a right triangle. This function represents the ratio of the adjacent side to the opposite side in a right triangle.

Mathematically, cotangent can be expressed in terms of the sine and cosine functions as follows:

• \( \cot x = \frac{\cos x}{\sin x} \)
• It can also be seen as the reciprocal of the tangent function: \( \cot x = \frac{1}{\tan x} \)

The cotangent function is particularly useful in trigonometry for simplifying expressions and solving equations. Understanding cotangent and its relationship with sine and cosine helps recognize and work with various fundamental trigonometric identities.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions (sine, cosine, and tangent) by taking their reciprocals. They play an important role in simplifying trigonometric expressions and solving equations.

The three common reciprocal functions are:

• Cosecant (csc): This is the reciprocal of sine. \( \csc x = \frac{1}{\sin x} \)
• Secant (sec): This is the reciprocal of cosine. \( \sec x = \frac{1}{\cos x} \)
• Cotangent (cot): As mentioned previously, this is the reciprocal of tangent. \( \cot x = \frac{1}{\tan x} \)

These reciprocal relationships can often simplify complex trigonometric expressions, allowing for easier manipulation. Recognizing reciprocal functions and using them effectively is a key skill in trigonometry.

Fundamental Identities

Fundamental identities in trigonometry are equations involving trigonometric functions that are true for every value of the variables where the functions are defined. Knowing these identities is useful for simplifying and solving trigonometric expressions.

Some of the key fundamental identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Relationship Between Cotangent and Tangent: \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)
• For secant and cosecant: \( \sec^2 x = 1 + \tan^2 x \) and \( \csc^2 x = 1 + \cot^2 x \)

These identities allow for converting one function into another, simplifying expressions, and solving trigonometric equations. Mastering these identities is crucial for solving a wide variety of problems in trigonometry, such as verifying the match \( \frac{\cos x}{\sin x} = \cot x \) with a fundamental identity.