Problem 85 - Algebra 2

Question

Simplify each expression. $$ \frac{\cos \theta \csc \theta}{\cot \theta} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sin \theta \).

Step 1: Express in terms of sine and cosine

Express every term in terms of sine and cosine. This changes the expression \( \frac{\cos \theta \csc \theta}{\cot \theta} \) to \( \frac{\cos \theta}{\sin \theta}}{\cos \theta / \sin \theta} \).

Step 2: Simplify fractions

The next step is to simplify this expression by multiplying the numerator and denominator by \( \sin \theta \). Now we have \( \frac{\cos \theta \sin \theta}{\cos \theta} \).

Step 3: Cancel common factors

Our last step is to cancel out any term appearing in both numerator and denominator. The \( \cos \theta \) cancels out, leaving us with \( \sin \theta \).

Key Concepts

Simplifying Expressions

When working with mathematical problems, simplifying expressions makes them easier to understand. It often involves reducing a complex expression into a simpler form.
Simplifying helps in solving problems efficiently and revealing underlying mathematical properties. Let's explore how to simplify an expression using the example \( \frac{\cos \theta \csc \theta}{\cot \theta} \).

• Convert all terms into basic elements: Here, we start by expressing each trigonometric function in terms of sine and cosine.
• Simplify by reducing fractions: Combine terms and reduce expressions, usually by multiplying both numerator and denominator by the same value. This often makes canceling possible.
• Cancel common terms: In our example, \( \cos \theta \) is canceled out, leaving you with a final simple expression, \( \sin \theta \).

By following these steps, complex trigonometric expressions become straightforward. This method can be applied to various mathematical challenges to make them more manageable.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions, essential for understanding and manipulating trigonometric expressions.
These functions relate to angles in a right triangle or a unit circle. By expressing other functions in terms of sine and cosine, simplifying becomes much easier.

• **Sine**: Represented as \( \sin \theta \), it is the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine**: Represented as \( \cos \theta \), it is the ratio of the adjacent side to the hypotenuse.

This relationship helps transform complex expressions. For the given exercise, by converting \( \csc \theta \) and \( \cot \theta \) into these basic functions:
\( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), the expression becomes simpler to handle and simplify.

Trigonometric Functions

Trigonometric functions are not just about angles and triangles. They are tools that help in various fields like physics, engineering, and signal processing.
In math, they serve as the foundation for working with periodic phenomena and transforming complex numbers.

• **Basic Functions**: Includes sine, cosine, tangent, cotangent, secant, and cosecant.
• **Relationships**: Understanding how these functions relate to each other helps in simplifying and solving trigonometric equations.
• **Applications**: Used to analyze wave patterns, model oscillations, and solve real-world problems like sound, light, and tides.

In the context of simplifying expressions, trigonometric functions and their relationships allow you to convert and reduce expressions effectively. This understanding leads to a stronger grasp of mathematics and its applications in various domains.