Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\frac{\tan 80^{\circ}+\tan 55^{\circ}}{1-\tan 80^{\circ} \tan 55^{\circ}}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(-1\).

Step 1: Identify the Trigonometric Identity

This expression is of the form \(\frac{\tan A + \tan B}{1 - \tan A \tan B}\), which is the tangent addition formula: \(\tan(A + B)\).

Step 2: Calculate the Angle Sum

Using the identity, identify that \(A = 80^{\circ}\) and \(B = 55^{\circ}\). Calculate \(A + B = 80^{\circ} + 55^{\circ} = 135^{\circ}\).

Step 3: Find the Exact Value of the Tangent

The tangent of \(135^{\circ}\) can be found using the unit circle. Since \(135^{\circ}\) is in the second quadrant where tangent is negative, and the reference angle is \(45^{\circ}\), \(\tan 135^{\circ} = -\tan 45^{\circ} = -1\). Thus, the expression simplifies to \(-1\).

Key Concepts

Tangent Addition Formula

The tangent addition formula is a powerful tool in trigonometry. It allows us to find the tangent of the sum of two angles using the formula: \[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\] This formula is particularly useful when we know or can easily find the tangents of the individual angles, but it might be challenging to work directly with the sum. Whether you’re tackling geometric problems or simplifying trigonometric expressions, recognizing the structure of this formula can be invaluable.

• The numerator consists of the direct sum of the tangents: \(\tan A + \tan B\).
• The denominator accounts for the interaction between the two tangents through their product: \(1 - \tan A \tan B\).

Understanding this setup can simplify many otherwise complex calculations, allowing you to transform a seemingly difficult problem into a straightforward evaluation of a single tangent.

Angle Sum

Calculating the angle sum is usually the first concrete step after identifying a trigonometric identity that involves addition or subtraction. Here, we focus on adding the calculated or given angles together. In our example, we need to sum \(A = 80^{\circ}\) and \(B = 55^{\circ}\) together:\[A + B = 80^{\circ} + 55^{\circ} = 135^{\circ}\]This process is straightforward:

• Add the degrees directly.
• Ensure that the result is interpreted correctly based on the trigonometric quadrant rules. Revise which quadrant the sum places you in.

Simple sums can lead to complex conclusions when related to different properties of trigonometric functions such as signs or periodicity, which influence their values relative to specific angles.

Exact Value Calculation

Finding the exact value of trigonometric functions often involves understanding the unit circle and associated reference angles. In our example, the target angle is \(135^{\circ}\). To find \(\tan 135^{\circ}\), follow these steps:

• Recognize that \(135^{\circ}\) lies in the second quadrant.
• Conclude that in the second quadrant, the tangent function is negative since tangent is positive in the first and third quadrants.
• Identify the reference angle, which is \(\,45^{\circ}\), because \(135-90=45\).
• Use the tangent of this reference angle: \(\tan 45^{\circ} = 1\).
• Apply the quadrant sign rule: Since \(135^{\circ}\) is in the second quadrant, \(\tan 135^{\circ} = -1\).

With these steps, trigonometric problems become a puzzle, where you can accurately predict outcomes based on understanding foundational unit circle concepts and quadrant rules.