Problem 19 - Algebra and Trigonometry

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 expression simplifies to \( \sqrt{3} \).

Step 1: Identify the Formula

The given expression \( \frac{\tan 73^{\circ}-\tan 13^{\circ}}{1+\tan 73^{\circ} \tan 13^{\circ}} \) fits the subtraction identity for tangent: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \). This means we can use this identity to simplify the expression to \( \tan(73^{\circ} - 13^{\circ}) \).

Step 2: Calculate the Angle Difference

Now, subtract the angles inside the tangent function: \( 73^{\circ} - 13^{\circ} \). This simplifies to \( 60^{\circ} \).

Step 3: Use Known Values

We know the exact trigonometric value for \( \tan(60^{\circ}) \), which is \( \sqrt{3} \). Thus, the expression simplifies to \( \sqrt{3} \).

Key Concepts

Tangent Subtraction Formula

When dealing with expressions involving the tangent of angle differences, the tangent subtraction formula comes to our rescue. The formula is: \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \] This formula is essential when simplifying trigonometric expressions where subtracting angles is involved.

• Effectively Simplifies Expressions: It transforms a complex tangent difference into something more manageable, a single tangent of a simplified angle.
• Useful in Various Applications: Be it in solving trigonometric equations or integrating trigonometry into calculus, this identity is quite effective.

With this formula, a complex-looking fraction can often reduce to a simple tangent function of a single angle.

Angle Difference Identity

The angle difference identity is a helpful tool for simplifying trigonometric functions that involve two angles. Specifically, it helps us express tangents of differences in angles concisely. In the context of the tangent subtraction formula, the angle difference identity simplifies calculations by reducing them to operations with single angles. This is why we used it in our problem: to condense the expression to \( \tan(73^{\circ} - 13^{\circ}) \), which simplifies further to \( \tan(60^{\circ}) \).

• Makes Complex Calculations Simpler: By allowing calculations to occur with single angles instead of multiple ones, the identity significantly simplifies the process.
• Foundation for Applications: Many applications in physics and engineering rely on this identity for derivations and solving complex angle-related problems.

Understanding this identity allows for efficient computations in trigonometric problems.

Exact Trigonometric Values

Exact trigonometric values play a critical role in trigonometry, as they provide precise results for commonly encountered angles. These values are often relied upon to simplify expressions quickly without resorting to a calculator. Common exact values to remember include:

• For \( \tan(45^{\circ}) = 1\): This is based on the idea of an isosceles right triangle.
• For \( \tan(60^{\circ}) = \sqrt{3} \): Often used in expressions involving 30-60-90 triangles.
• For \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \): Complements the 60-degree measure in 30-60-90 triangles.

Being familiar with these values means you can solve trigonometric problems much more efficiently by recognizing exact values in angles like \( 60^{\circ} \) immediately. In our problem, recognizing \( \tan(60^{\circ}) \) as \( \sqrt{3} \) allowed us to find the solution almost instantaneously.