Problem 30 - Algebra 2

Question

Simplify each trigonometric expression. $$ \frac{\tan \theta}{\sec \theta-\cos \theta} $$

Step-by-Step Solution

Answer

The simplified form of the trigonometric expression \(\frac{\tan \theta}{\sec \theta-\cos \theta}\) is \(\frac{1}{\cos \theta}\)

Step 1: Rewrite secant in terms of cosine

The first step is to rewrite \(\sec \theta\) in terms of \(\cos \theta\). This gives us the expression \(\frac{\tan \theta}{\frac{1}{\cos \theta}-\cos \theta}\).

Step 2: Simplify the denominator

The next step is to simplify the denominator of the expression by finding a common denominator. This results in \(\frac{\tan \theta}{1-\cos^2 \theta}\).

Step 3: Use the identity for tangent

In this step, the identity for tangent, which is \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), is used. The expression becomes \(\frac{\frac{\sin \theta}{\cos \theta}}{1-\cos^2 \theta}\).

Step 4: Apply the Pythagorean identity and simplify

To simplify further, apply the Pythagorean identity \(1-\cos^2 \theta = \sin^2 \theta\). This changes the expression to \(\frac{\frac{\sin \theta}{\cos \theta}}{\sin^2 \theta}\), or simply \(\frac{1}{\cos \theta}\).

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions can sometimes seem daunting, but by breaking down each component, it becomes more manageable. The key is in understanding how to manipulate different trigonometric functions to reduce an expression to its simplest form. Let's take the given expression: \( \frac{\tan \theta}{\sec \theta-\cos \theta} \). To simplify, start by expressing all trigonometric terms in a common form. We rewrite \( \sec \theta \) in terms of cosine, yielding \( \frac{1}{\cos \theta} \). This turns the expression into \( \frac{\tan \theta}{\frac{1}{\cos \theta} - \cos \theta} \). From here, it's a matter of algebraic manipulation and applying familiar identities in a step-by-step manner. Next, focus on simplifying the denominator by finding a common base. This results in \( \frac{\tan \theta}{1 - \cos^2 \theta} \). As you can see, having a firm grasp of working with algebraic fractions and converting trigonometric functions into simpler terms is crucial when simplifying complex expressions.

Pythagorean Identity

The Pythagorean identity is one of the most vital tools in trigonometry. It expresses the fundamental relationship between basic trigonometric functions, particularly sine and cosine. The identity states: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This powerful equation allows us to substitute parts of an expression to simplify or even solve it. In the exercise, during the simplification, we see the term \( 1 - \cos^2 \theta \). By applying the Pythagorean identity, we can replace it with \( \sin^2 \theta \). Thus, the expression becomes \( \frac{\frac{\sin \theta}{\cos \theta}}{\sin^2 \theta} \). This substitution step is key in reducing the complexity of the equation and demonstrates how these identities serve as shortcuts for intricate calculations. Using such trigonometric identities alleviates the need for longer calculations and aids in developing a deeper understanding of how these functions interrelate.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are the building blocks of trigonometry. Understanding their relationships and how to express them in terms of each other is essential for simplifying and solving trigonometric expressions.Here's a brief overview of these functions:

• Sine (\( \sin \theta \)): Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos \theta \)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): Represents the ratio of the sine to the cosine, i.e., \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Secant (\( \sec \theta \)): Is the reciprocal of cosine, i.e., \( \sec \theta = \frac{1}{\cos \theta} \).

The exercise involves transforming these functions and using their identities to deduce a simpler form. Knowing how to switch between these forms—and being comfortable with their properties—is a skill that enhances your ability to tackle even the most complex trigonometric problems with ease.