Problem 30 - Precalculus with Limits

Question

In Exercises 25 - 30, match the trigonometric expression with one of the following. (a)\( \sec x \) (b) \( -1 \) (c) \( \cot x \) (d) \( 1 \) (e) \( -\tan x \) (d) \( \sin x \) \( \dfrac{\sin\left[(\pi/2) - x\right]}{\cos\left[(\pi/2) - x\right]} \)

Step-by-Step Solution

Answer

\(\cot x\)

Step 1: Identify the Co-Function Identities

The first step is to correctly identify the trigonometric functions within the expression. Here we have \(\sin(\pi/2 - x)\) and \(\cos(\pi/2 - x)\). According to the co-function identities, these are equivalent to \(\cos(x)\) and \(\sin(x)\) respectively.

Step 2: Apply the Co-Function Identities

Next, we substitute the identities into the expression. So, \(\sin(\pi/2 - x) / \cos(\pi/2 - x)\) becomes \(\cos(x) / \sin(x)\).

Step 3: Simplify the Expression

The expression \(\cos(x) / \sin(x)\) is equivalent to \(cot(x)\). This is one of the standard trigonometric identities which states that \(cot(x) = cos(x) / sin(x)\).

Key Concepts

Co-function Identities

Trigonometric functions have a unique set of relationships known as co-function identities. These identities establish a link between complementary angles. An angle is complementary to another if their sum equals \( 90^{\circ} \) or \( \pi/2 \) radians.

Co-function identities specifically show how sine and cosine, or tangent and cotangent, relate to each other when their angles are complementary. For example:

• \( \sin(\pi/2 - x) = \cos(x) \)
• \( \cos(\pi/2 - x) = \sin(x) \)

These identities help simplify expressions involving angles, making calculations easier. In our given exercise, noticing these identities allows us to simplify the expression \( \dfrac{\sin((\pi/2) - x)}{\cos((\pi/2) - x)} \) into something more recognizable, eventually identifying it as a cotangent function.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves recognizing and applying identities to rewrite the expression in a simpler form. The primary goal is to make the expression easier to work with or comparable to other expressions. Here's a brief roadmap to help simplify trigonometric expressions:

• **Identify Known Identities**: Use standard identities, like the Pythagorean identities or co-function identities, to recognize equivalent simpler forms.
• **Substitution**: Once the identity is identified, substitute the known equivalents into the expression.
• **Reduce and Interpret**: Simplify fractions or expressions to make them interpretable as standard functions.

In our problem, \( \sin((\pi/2) - x) \) and \( \cos((\pi/2) - x) \) were substituted with \( \cos(x) \) and \( \sin(x) \) respectively, using co-function identities. This transformed the expression into \( \cos(x) / \sin(x) \), which is simpler and more coordinated with recognized trigonometric functions.

Cotangent Function

The cotangent function is one of the fundamental trigonometric functions. It is defined as the reciprocal of tangent: \[ \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \]

Understanding the cotangent function involves recognizing its relationship with sine and cosine. Essentially, it represents the ratio of the cosine of an angle to the sine of that angle.

The cotangent is useful in various applications, particularly when dealing with right triangles and periodic functions in trigonometry. In our exercise, the final simplified expression \( \dfrac{\cos(x)}{\sin(x)} \) is the cotangent function \( \cot(x) \). Recognizing this helps in matching the expression to a standard list of trigonometric functions, confirming our solution.