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. \(\mathbf{I}\) \(\frac{\cos x}{\sin x}=\)_______ \(\mathbf{II}\) 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

Column I expression \(\frac{\cos x}{\sin x}\) matches option B: cot \(x\).

Step 1: Understand the Expression in Column I

The expression given in Column I is \( \frac{\cos x}{\sin x} \). This expression represents a trigonometric function which is the ratio of cosine to sine.

Step 2: Recall Trigonometric Identities

Recall that the trigonometric identity for cotangent is \( \cot x = \frac{\cos x}{\sin x} \). This means cotangent is the function that is the reciprocal of tangent, which is \( \tan x = \frac{\sin x}{\cos x} \).

Step 3: Match with Options in Column II

In Column II, option B matches the identity \( \cot x = \frac{\cos x}{\sin x} \). None of the other options represent this identity. Option D is \( \frac{\sin x}{\cos x} \), which is tangent, not cotangent.

Step 4: Verify the Correctness of Option B

Double-check that \( \frac{\cos x}{\sin x} = \cot x \) to ensure it correctly matches the standard trigonometric identities. This confirms that it is a correct match for the expression in Column I.

Key Concepts

Cotangent

Cotangent is one of the lesser-known trigonometric functions, which often causes confusion among students. It is closely related to the tangent function. In essence, cotangent is the reciprocal of the tangent function. The cotangent (often abbreviated as \( \cot \)) is defined as the ratio of the cosine of an angle to the sine of the same angle.

This can be expressed as:

• \( \cot x = \frac{\cos x}{\sin x} \)

In a right-angled triangle, this would mean that cotangent represents the ratio of the length of the adjacent side to the opposite side. It is important to remember that since cotangent is the reciprocal of tangent, when the tangent is zero or undefined (as the sine function approaches zero), the cotangent will be undefined or zero respectively.

Understanding cotangent helps in solving various trigonometric equations and identities efficiently. When you see an expression of \( \frac{\cos x}{\sin x} \), it is essential to recognize it as cotangent for quick identification and problem-solving.

Sine and Cosine Ratios

Sine and cosine are fundamental trigonometric ratios, critical for defining other trigonometric functions and identities. They are derived from the right triangle and the unit circle definitions. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Cosine is similar but refers to the ratio of the adjacent side to the hypotenuse.

These can be formulated as:

• Sine: \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine: \( \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} \)

On the unit circle, where the hypotenuse is always 1, the sine of an angle is simply the y-coordinate, and the cosine is the x-coordinate of the point where the terminal side of the angle intersects the circle. This understanding supports the connection between the sine and cosine functions and their complementary relationship in trigonometric identities.

Deep comprehension of sine and cosine allows you to derive many trigonometric identities, such as \( \sin^2 x + \cos^2 x = 1 \), which is a foundational identity in trigonometry.

Tangent Function

The tangent function is another core trigonometric ratio and is often the first 'complex' function students learn after sine and cosine. The tangent of an angle, denoted \( \tan x \), is the ratio of the sine of the angle to the cosine of the angle. This means it can be expressed as:

• \( \tan x = \frac{\sin x}{\cos x} \)

In the context of a right triangle, the tangent is the ratio of the length of the opposite side to the adjacent side. It's important to realize that tangent, having sine and cosine in its ratio, is undefined when the cosine of the angle is zero (leading to division by zero). This typically occurs at angles of 90° plus multiples of 180° (or \( \frac{\pi}{2} \) plus multiples of \( \pi \) radians).

The tangent function is periodic, with a period of \( \pi \) radians, meaning it repeats its values every 180°. Understanding the behavior of tangent helps students in graphing and solving trigonometric equations that involve angles varying across a range of radians or degrees.