Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\tan (A+B)\) and of \(\tan (A-B)\) for each given pair of values. $$ A=360^{\circ}, B=60^{\circ} $$

Step-by-Step Solution

Answer

\( \tan(420^{\circ}) = \sqrt{3} \), \( \tan(300^{\circ}) = -\sqrt{3} \).

Step 1: Understanding Tangent Addition Formula

We need to find the value of \( \tan(A + B) \) and \( \tan(A - B) \). The formula for tangent of a sum is \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \) and for a difference \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B} \).

Step 2: Calculating Tangents for Specific Angles

Calculate \( \tan(360^{\circ}) \) and \( \tan(60^{\circ}) \). We know \( \tan(360^{\circ}) = 0 \) because it is a full circle, and \( \tan(60^{\circ}) = \sqrt{3} \).

Step 3: Applying Tangent Addition Formula

Using \( \tan A = 0 \) and \( \tan B = \sqrt{3} \), substitute into the sum formula: \[ \tan(360^{\circ} + 60^{\circ}) = \frac{0 + \sqrt{3}}{1 - 0 \cdot \sqrt{3}} = \sqrt{3}. \]

Step 4: Applying Tangent Subtraction Formula

Using \( \tan A = 0 \) and \( \tan B = \sqrt{3} \), substitute into the difference formula: \[ \tan(360^{\circ} - 60^{\circ}) = \frac{0 - \sqrt{3}}{1 + 0 \cdot \sqrt{3}} = -\sqrt{3}. \]

Key Concepts

Understanding the Tangent Addition Formula

The tangent addition formula is a handy tool in trigonometry for finding the tangent of the sum of two angles. This formula is written as:

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

It simplifies the calculation of the tangent of the sum by breaking it into a more manageable form using individual tangents.
Here's how it works:

• \( \tan A \) and \( \tan B \) are the individual tangents of angles \( A \) and \( B \).
• The numerator \( \tan A + \tan B \) is the sum of these tangents.
• The denominator \( 1 - \tan A \cdot \tan B \) carefully accounts for the interaction between the two original tangents.

By substituting specific values for \( A \) and \( B \), like in this exercise where \( A = 360^{\circ} \) and \( B = 60^{\circ} \), you can calculate the exact result easily. You input these known tangents into the formula to find \( \tan(A + B) \).
This process is straightforward, especially when you know the standard tangent values for common angles.

Decoding the Tangent Subtraction Formula

The tangent subtraction formula is quite similar to the addition formula. It helps calculate the tangent of the difference between two angles. Its formula is:

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

Just like the addition formula, it is designed to break down the problem into simpler parts:

• Start by finding the individual tangents \( \tan A \) and \( \tan B \).
• Substitute these into the formula where the numerator becomes \( \tan A - \tan B \).
• The denominator changes slightly to \( 1 + \tan A \cdot \tan B \) to accommodate the subtraction.

In the exercise with \( A = 360^{\circ} \) and \( B = 60^{\circ} \), applying the formula helps find \( \tan(A - B) \). Knowing the tangents of standard angles, like \( \tan(60^{\circ}) = \sqrt{3} \) and \( \tan(360^{\circ}) = 0 \), simplifies the solution.
With precise substitution, this formula shows how differences affect tangents.

Exploring Specific Angle Tangents

Understanding specific angle tangents is crucial for using tangent formulas effectively. Some angles have well-known tangent values that are frequently used in trigonometry:

• \( \tan(0^{\circ}) = 0 \)
• \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \)
• \( \tan(45^{\circ}) = 1 \)
• \( \tan(60^{\circ}) = \sqrt{3} \)
• \( \tan(90^{\circ}) \) is undefined
• \( \tan(180^{\circ}) = 0 \)
• \( \tan(360^{\circ}) = 0 \)

When tackling an exercise, recognizing these values becomes essential. They not only make the math simpler but also allow you to solve tangent problems quickly.
For example, in our exercise:- Knowing \( \tan(60^{\circ}) = \sqrt{3} \) provided a crucial step in solving both the tangent addition and subtraction problems.- Recognizing \( \tan(360^{\circ}) = 0 \) helped in simplifying calculations further.
By memorizing these values, you gain a better grasp of how angles interact with tangents, making your trigonometry skills sharp and efficient.