Problem 55 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \sin \beta \tan \beta + \cos \beta \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \sin \beta \tan \beta + \cos \beta \) is \( \sec \beta \).

Step 1: Substitute using the quotient identity

First, using the quotient identity \( \tan \beta = \sin \beta/ \cos \beta \), at the place of \( \tan \beta \) substitute \( \sin \beta/ \cos \beta \). So our equation becomes \( \sin \beta (\sin \beta/ \cos \beta) + \cos \beta \), which simplifies to \( (\sin^2 \beta/ \cos \beta) + \cos \beta \).

Step 2: Find common denominators and simplify

Next, simplify by finding common denominators. Multiply the second term by \( \cos \beta/ \cos \beta \) to get \( (\sin^2 \beta + \cos^2 \beta)/ \cos \beta \)

Step 3: Substitute using the Pythagorean Identity

Finally, using the Pythagorean Identity \( \sin^2 \beta + \cos^2 \beta =1 \), substitute \( 1 \) for (\( \sin^2 \beta + \cos^2 \beta \)). The expression simplifies to \( 1/ \cos \beta \), which is equivalent to \( \sec \beta \) because \( \sec \beta = 1/\cos \beta \).

Key Concepts

Quotient Identity

Let's explore the Quotient Identity, a basic building block in trigonometry that helps simplify expressions. This identity states that:

• \( \tan \beta = \frac{\sin \beta}{\cos \beta} \)

What does this mean? \( \tan \beta \) is essentially the ratio of \( \sin \beta \) to \( \cos \beta \). This is important because it allows us to express tangent in terms of sine and cosine, the two primary trigonometric functions.
By converting \( \tan \beta \) to \( \frac{\sin \beta}{\cos \beta} \), we are able to manipulate and simplify expressions that contain tan, using just sine and cosine.

In our exercise, this substitution was the first step of simplification. By switching \( \tan \beta \) to \( \frac{\sin \beta}{\cos \beta} \), we transformed the expression, making it easier to work with later steps. This identity helps break down complex trigonometric expressions into more manageable parts.

Pythagorean Identity

Another cornerstone in trigonometry is the Pythagorean Identity, which states:

• \( \sin^2 \beta + \cos^2 \beta = 1 \)

This identity reveals the fundamental relationship between sine and cosine. No matter the angle \( \beta \), the sum of the squares of \( \sin \beta \) and \( \cos \beta \) will always equal one. This might remind you of the Pythagorean theorem. That's because trigonometric functions are deeply tied to geometry and the unit circle.

In the exercise, after substituting the Quotient Identity, we arrived at an expression: \( \frac{\sin^2 \beta + \cos^2 \beta}{\cos \beta} \). Here, thanks to the Pythagorean Identity, we can replace \( \sin^2 \beta + \cos^2 \beta \) with 1. This is crucial for simplifying trigonometric expressions to reach a more straightforward form, allowing us to see relations between different functions.

Simplifying Expressions

Simplifying expressions is a common task in mathematics that helps you turn complex equations into simpler forms. In trigonometry, this involves using identities like the Quotient and Pythagorean identities.

• Start by identifying key trigonometric identities you can use to substitute parts of the expression.
• Next, work to combine and manipulate terms to create a common denominator if necessary.
• Then, apply known identities to simplify.

In the given exercise, it started with converting \( \tan \beta \) using the Quotient Identity. The resulting expression \( \frac{\sin^2 \beta}{\cos \beta} + \cos \beta \) was further simplified by finding a common denominator. Finally, the Pythagorean Identity substituted \( \sin^2 \beta + \cos^2 \beta \) with 1, leading to \( \frac{1}{\cos \beta} \). This shows simplifying isn't just about making it shorter; it's about making it clearer to work with. Connecting and using these identities efficiently is key to mastering trigonometry.