Problem 14 - 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. \(\mathbf{I}\) \(\tan x=\)_______ \(\mathbf{II}\) 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

In Column II, option D (\( \frac{\sin x}{\cos x} \)) completes the identity for \( \tan x \).

Step 1: Identify the Expression for Tangent

In trigonometry, the tangent function is defined as the ratio of the sine and cosine functions. Therefore, \( \tan x \) is given by the expression \( \frac{\sin x}{\cos x} \). This matches one of the options in Column II.

Step 2: Match the Expression

From the options given in Column II, identify which option represents \( \tan x \). The option D, \( \frac{\sin x}{\cos x} \), matches the expression for \( \tan x \) derived in the previous step.

Key Concepts

Tangent Function

The tangent function, commonly denoted as \( \tan x \), plays a crucial role in trigonometry. It is the ratio of the sine function \( \sin x \) to the cosine function \( \cos x \). This means that the formula for tangent is given by: \[ \tan x = \frac{\sin x}{\cos x} \] This identity helps us understand the tangent's relation to the more basic trigonometric functions, sine and cosine.

• The tangent function is periodic with a period of \( \pi \) radians, which means it repeats its values every \( 180^\circ \) or \( \pi \) radians.
• It is undefined for angles where \( \cos x = 0 \), such as \( \frac{\pi}{2} + n\pi \) where \( n \) is an integer, due to division by zero.

Understanding this basic ratio helps simplify many trigonometric problems since all other functions can, in one way or another, be tied back to sine and cosine.

Sine and Cosine Functions

Sine and cosine are perhaps the most fundamental trigonometric functions. They are defined on the unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane. Here's a brief breakdown of each:

• Sine Function \( \sin x \): Represents the vertical coordinate of a point on the unit circle corresponding to an angle \( x \).
• Cosine Function \( \cos x \): Represents the horizontal coordinate of a point on the unit circle corresponding to an angle \( x \).

These functions have period \( 2\pi \), meaning they complete one full cycle every \( 360^\circ \) or \( 2\pi \) radians. The sine and cosine functions are related through several identities, one of the most prominent being the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] This identity is key in solving many trigonometric equations and demonstrates how the lengths of the sides of a right triangle relate to the hypotenuse. It also forms the backbone of many other trigonometric identities.

Fundamental Identities

Fundamental trigonometric identities are essential tools in solving a wide variety of problems. These identities reveal how the sine, cosine, and other trigonometric functions interrelate. Some of the most significant fundamental identities include:

• Pythagorean Identities: These relate the square of sine and cosine functions: \[ \sin^2 x + \cos^2 x = 1 \]
• Reciprocal Identities: They express trigonometric functions as reciprocals of each other: \[ \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{1}{\tan x} \]
• Quotient Identities: These identities express tangent and cotangent in terms of sine and cosine: \[ \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x} \]

These identities are indispensable in all fields requiring trigonometry, from geometry and calculus to engineering and physics. By mastering these, students gain a powerful toolkit for simplifying complex problems, verifying equations, and proving more advanced trigonometric statements.