Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-22,\) write each given expression in terms of sine and cosine and express the result in simplest form. \((\sec \theta)(\cot \theta)\)

Step-by-Step Solution

Answer

\((\sec \theta)(\cot \theta) = \frac{1}{\sin \theta}\).

Step 1: Express Secant in Terms of Cosine

The secant function, \( \sec \theta \), is defined as the reciprocal of the cosine function. Therefore, we can write it as \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Express Cotangent in Terms of Sine and Cosine

The cotangent function, \( \cot \theta \), is the reciprocal of the tangent function. Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we have \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 3: Multiply the Expressions

Now we multiply \( \sec \theta = \frac{1}{\cos \theta} \) by \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).\[\left( \frac{1}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right) = \frac{1 \cdot \cos \theta}{\cos \theta \cdot \sin \theta} \]

Step 4: Simplify the Expression

The \( \cos \theta \) terms in the numerator and denominator cancel each other out. Therefore, we obtain: \[\frac{1}{\sin \theta}\] This is the simplification of the given expression in terms of sine.

Key Concepts

Sine and Cosine

To understand the basis of trigonometric identities, especially when simplifying expressions like \((\sec \theta)(\cot \theta)\), it's crucial to understand sine and cosine. These functions form the foundation of trigonometry.

• Sine (\(\sin \theta\)): This function relates the angle \(\theta\) in a right triangle to the ratio of the length of the side opposite this angle over the hypotenuse.
• Cosine (\(\cos \theta\)): Similarly, this is the ratio of the length of the adjacent side to angle \(\theta\) over the hypotenuse.

These trigonometric functions are fundamental because they allow us to convert other trigonometric functions into forms that are commonly easier to manipulate. This is particularly useful when dealing with complex trigonometric expressions. When we rewrite functions in terms of sine and cosine, like secant and cotangent such as in the original problem, we're leveraging these basic identities to simplify expressions efficiently.

Trigonometric Functions

Trigonometric functions extend beyond just sine and cosine. They include various other functions such as tangent, cotangent, secant, and cosecant. Understanding how these relate and interact is essential for expression manipulation and simplification.

• Tangent (\(\tan \theta\)): Defined as the ratio of sine to cosine \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). It highlights the relationship between sine and cosine directly.
• Cotangent (\(\cot \theta\)): The reciprocal of tangent, \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\). This identity is pivotal in converting \(\cot \theta\) to terms of sine and cosine, as seen in the exercise.
• Secant (\(\sec \theta\)): A fundamental reciprocal identity where \(\sec \theta = \frac{1}{\cos \theta}\). This is used in the exercise to change \(\sec \theta\) into a cosine-based form.
• Cosecant (\(\csc \theta\)): The reciprocal of sine, \(\csc \theta = \frac{1}{\sin \theta}\).

Each function provides a way to bridge angles and sides in triangles, offering multiple entry points to solve or simplify equations by converting less familiar functions into terms using sine and cosine.

Simplification of Expressions

Simplifying trigonometric expressions often relies on converting all parts of the equation into sine and cosine. Let's dive into how this process works, using the original step by step solution as an example.

The expression given was \((\sec \theta)(\cot \theta)\). By recalling the definitions:

• Secant: \(\sec \theta = \frac{1}{\cos \theta}\)
• Cotangent: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

By substituting these identities into the expression, it becomes manageable:

• We multiply: \(\left( \frac{1}{\cos \theta} \right) \left( \frac{\cos \theta}{\sin \theta} \right)\).
• When multiplied, the \(\cos \theta\) terms cancel, leaving \(\frac{1}{\sin \theta}\).

This completes the simplification process, arriving at a form expressed solely in terms of sine, which is easier to interpret. Simplifying expressions in trigonometry is about knowing these identities and fluently applying them to reduce complexity.