Problem 38 - 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. \( \cos \beta \tan \beta \)

Step-by-Step Solution

Answer

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

Step 1: Identify the appropriate trigonometric identity

Identify the trigonometric identity that can simplify the given expression. In this case, the identity that fits is \(\tan \beta = \frac{\sin \beta}{\cos \beta}\)

Step 2: Substitute the identity into the expression

Substitute \(\tan \beta\) in the original expression \( \cos \beta \tan \beta \) with \(\frac{\sin \beta}{\cos \beta}\), to get \( \cos \beta \cdot \frac{\sin \beta}{\cos \beta} \)

Step 3: Simplify the expression

The \( \cos \beta \) in the numerator and the denominator cancel out, leaving the simplified expression as \( \sin \beta \)

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying trigonometric expressions. These identities are like formulas that relate different trigonometric functions to one another. There are a few fundamental identities that are often used, such as:

• Reciprocal identities, e.g., \ \( \tan \beta = \frac{\sin \beta}{\cos \beta} \)
• Pythagorean identities, e.g., \ \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Quotient identities, which express one trigonometric function as a quotient of two others

Using these identities helps to simplify complex trigonometric expressions by revealing equivalent, often simpler forms. Understanding and remembering these identities is crucial when working with trigonometric problems. They are the foundation blocks of solving equations and simplifying expressions in trigonometry.

Simplification of Expressions

Simplifying trigonometric expressions involves using identities to transform a given expression into its simplest form. This process can significantly reduce the complexity of calculations.

• Identify the relevant identity that can be applied
• Rewrite the expression using this identity
• Perform mathematical operations such as cancellation

In the exercise given, the expression \ \( \cos \beta \tan \beta \) was simplified using the identity \ \( \tan \beta = \frac{\sin \beta}{\cos \beta} \).
The \ \( \cos \beta \) factors cancel out from the denominator and the numerator, leaving the simpler form \ \( \sin \beta \). This demonstrates how simplification makes expressions easier to manage.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent describe relationships in right triangles, connecting angles to side lengths. These functions extend beyond simple triangle problems, being critical in calculus, physics, and engineering.
They are defined as follows:

• \(\sin \theta\) is the opposite side over the hypotenuse
• \(\cos \theta\) is the adjacent side over the hypotenuse
• \(\tan \theta\) is the opposite side over the adjacent side

Trigonometric functions are periodic, which means they repeat their values over regular intervals.
This periodic nature allows for functions like \ \( \cos \beta \tan \beta \) to be expressed in various equivalent forms using identities. Understanding these functions allows us to tackle a wide array of mathematical and scientific problems.