Problem 13 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sec \theta \cot \theta $$

Step-by-Step Solution

Answer

After simplifying the given trigonometric expression, the result is \( \csc \theta \).

Step 1: Substitute Identities

Substitute \( \sec \theta \) with \( \frac{1}{\cos \theta} \) and \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \). This gives us the expression \( \frac{1}{\cos \theta} * \frac{\cos \theta}{\sin \theta} \).

Step 2: Simplify The Expression

The \( \cos \theta \) in both the numerator and denominator cancels out, which leads to the expression \( \frac{1}{\sin \theta} \).

Step 3: Convert to Traditional Trigonometric Function

The expression \( \frac{1}{\sin \theta} \) can be rewritten as \( \csc \theta \), which is the reciprocal of \( \sin \theta \).

Key Concepts

Trigonometric Simplification

Trigonometric simplification involves transforming an expression with trigonometric functions into a simpler form. The goal is to reduce complexity while maintaining equivalence. Simplification often relies on algebraic manipulation and substitution with basic trigonometric identities. In our example, we began by substituting the given trigonometric functions with their equivalent algebraic forms. This initial step sets the foundation for further simplification.

• Replacing the secant function: \( \sec \theta = \frac{1}{\cos \theta} \)
• Transforming the cotangent function: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

By substituting these identities, the expression becomes a fraction where like terms can cancel each other out. This process is crucial in reducing trigonometric expressions to a form that is easier to interpret or work with.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions by taking their reciprocal. These functions include cosecant, secant, and cotangent. They play a significant role in trigonometric simplification and transformation processes.

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

Recognizing these reciprocal functions and their relationships with sine, cosine, and tangent is vital for simplification tasks. In our exercise, identifying that \( \frac{1}{\sin \theta} \) is equal to \( \csc \theta \) is a key step in reaching a simplified form.

Cotangent and Secant Identities

Cotangent and secant are two important reciprocal trigonometric identities that frequently appear in various mathematical contexts. Familiarizing yourself with their identities allows for easier manipulation of trigonometric expressions.
The secant function, \( \sec \theta = \frac{1}{\cos \theta} \), measures the reciprocal of cosine and provides insights into the behavior of angles, especially those not possible in right-triangle contexts. Similarly, the cotangent function, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), combines both cosine and sine, showing the ratio between them.

• Secant relates to cosine directly as its inverse.
• Cotangent finds its use primarily in advanced mathematical applications due to its dual link with both sine and cosine.

In our specific exercise, using these identities as substitutions helped initially convert secant and cotangent into fractional forms, ultimately streamlining the expression into its simplest version as \( \csc \theta \). Mastery of these identities allows for more fluid transitions between complex expressions and their more manageable forms.