Problem 19 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. $$\frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}}$$

Step-by-Step Solution

Answer

The exact value is \( \sqrt{3} \).

Step 1: Identify the Tangent Subtraction Formula

The given expression \( \frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}} \) is in the form of the tangent subtraction identity: \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \] Recognize \( A = 73^{\circ} \) and \( B = 13^{\circ} \).

Step 2: Apply the Tangent Angle Subtraction Formula

Substitute \( A = 73^{\circ} \) and \( B = 13^{\circ} \) into the tangent subtraction formula. This results in: \[ \tan(73^{\circ} - 13^{\circ}) = \tan 60^{\circ} \] Simplify the angle inside the tangent function to get \( 60^{\circ} \).

Step 3: Calculate the Exact Value of the Tangent Function

The exact value of \( \tan 60^{\circ} \) is known from trigonometric tables: \( \tan 60^{\circ} = \sqrt{3} \). Therefore, the evaluation of the original expression as a single trigonometric function yields \( \sqrt{3} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics, allowing us to manipulate and simplify trigonometric expressions. They are equations involving trigonometric functions that are true for every value of the occurring variables. By using these identities, we can transform complex trigonometric expressions into simpler forms. This simplifies calculations and uncovers underlying relationships between different angles and functions.

One such identity is the Tangent Subtraction Formula, which is used to simplify the expression involving the difference of tangents. Specifically, it is given by:

• \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

This formula helps in re-writing the tangent of an angle difference as a single trigonometric function, facilitating easier computations. Using tangent identities with precision, as done in the given exercise, allows one to break down trigonometric problems into simpler parts that can be managed computationally, aiding in finding exact values.

Exact Value of Trigonometric Functions

The exact values of trigonometric functions are crucial in trigonometry, particularly when solving problems involving known angles. These values stem from the geometric relationships in a unit circle or an equilateral triangle, providing consistent outputs for angles expressed in degrees or radians.

For standard angles such as \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \), the trigonometric values are well-established and frequently memorized for quick reference:

• \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \)
• \( \tan 45^{\circ} = 1 \)
• \( \tan 60^{\circ} = \sqrt{3} \)

These exact values assist in efficiently solving trigonometric equations, as seen in our problem where the function reduces to \( \tan 60^{\circ} \). Recognizing and remembering these common values is particularly useful, as they allow for the quick determination of results without calculator support.

Angle Subtraction in Trigonometry

Angle subtraction in trigonometry involves subtracting one angle from another and then finding the trigonometric function of the resulting angle. This process is vital when dealing with expressions of trigonometric functions that involve angle differences. Such operations are commonly facilitated by the use of trigonometric identities for addition and subtraction.

In the given exercise, we worked with the difference between two angles, \( A = 73^{\circ} \) and \( B = 13^{\circ} \). By utilizing the Tangent Subtraction Formula, the expression was simplified to tan of a single angle:

• \( \tan(73^{\circ} - 13^{\circ}) = \tan 60^{\circ} \)

This shows how angle subtraction can simplify the evaluation of trigonometric expressions by reducing them to known angles with easily detectable exact values.

Mastering this concept is essential, as solving various trigonometric problems often necessitates working with angle subtraction and using it to access more straightforward computations or known trigonometric values.