Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each expression as a single term using \(\sin \theta, \cos \theta,\) or both. $$ \cot \theta \sec \theta $$

Step-by-Step Solution

Answer

The expression simplifies to \( \csc \theta \).

Step 1: Recall Trigonometric Identities

First, recall the definitions of the trigonometric functions involved. The cotangent of an angle \( \theta \) is defined as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \) and the secant of an angle \( \theta \) is defined as \( \sec \theta = \frac{1}{\cos \theta} \). We will use these identities to rewrite each term.

Step 2: Rewrite Each Function Using Sine and Cosine

Using the definitions from Step 1, rewrite \( \cot \theta \) and \( \sec \theta \) in terms of \( \sin \theta \) and \( \cos \theta \). We have:\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \]

Step 3: Multiply the Expressions

Next, multiply the expressions for \( \cot \theta \) and \( \sec \theta \):\[ \cot \theta \sec \theta = \left( \frac{\cos \theta}{\sin \theta} \right) \times \left( \frac{1}{\cos \theta} \right) \]

Step 4: Simplify the Expression

Simplify the product obtained in the previous step:\[ \cot \theta \sec \theta = \frac{\cos \theta}{\sin \theta} \times \frac{1}{\cos \theta} = \frac{\cos \theta \times 1}{\sin \theta \times \cos \theta} = \frac{1}{\sin \theta} = \csc \theta \]

Step 5: Write the Expression as a Single Term

The expression \( \cot \theta \sec \theta \) simplifies to \( \csc \theta \). Thus, the original expression is simplified to a single term using the cosecant function.

Key Concepts

Cotangent

The cotangent function, often abbreviated as \( \cot \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In terms of tangent, it is expressed as \( \cot \theta = \frac{1}{\tan \theta} \). Additionally, we can rewrite it using the sine and cosine functions:

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

This definition reveals that cotangent is essentially the ratio of the adjacent side to the opposite side in a right-angle triangle, especially focusing on the angle \( \theta \). It is an inverse function to the tangent and is widely used in various trigonometric transformations. Remember, when dealing with the cotangent, it's crucial to recognize it as a ratio of these fundamental trigonometric functions to simplify expressions thoroughly and correctly.

Secant

The secant function, represented as \( \sec \theta \), is another primary trigonometric function. It is defined as the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In geometric terms, when considering a unit circle, the secant function can be understood as the line extending from the center of the circle to the tangent line that touches the circle, passing through the angle \( \theta \). Essentially, it measures the length of the hypotenuse over the adjacent side of an angle in a triangle. It's particularly useful for simplifying expressions that involve fractional trigonometric calculations, where demanding operations are needed to convert complex expressions into simpler forms. Understanding \( \sec \theta \) as being dependent on cosine aids in recognizing its transformations and its role in equations like the one we are discussing here.

Cosecant

The cosecant function is represented by \( \csc \theta \) and stands as the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Similar to the secant, this function measures the ratio of the hypotenuse to the opposite side within a right-angled triangle. It's instrumental when dealing with expressions involving sine, as it transforms them by offering a simpler form to work with. Cosecant is especially noteworthy when simplifying products of trigonometric expressions, like in our exercise. When we simplify \( \cot \theta \sec \theta \) and find it equals \( \csc \theta \), it beautifully demonstrates how these trigonometric identities come together to simplify expressions using reciprocal relationships. Understanding and using \( \csc \theta \) aids in reorganizing trigonometric expressions efficiently.