Problem 10 - 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. $$ A=120^{\circ}, B=45^{\circ} $$

Step-by-Step Solution

Answer

\( \tan(120^{\circ} + 45^{\circ}) = -\sqrt{3} \), \( \tan(120^{\circ} - 45^{\circ}) = 1 + \sqrt{3} \).

Step 1: Recall Tangent Addition and Subtraction Formulas

For the sum of angles, the formula is \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). For the difference, it is \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).

Step 2: Find \( \tan 120^{\circ} \) and \( \tan 45^{\circ} \)

Recall that \( \tan 120^{\circ} = \tan(180^{\circ} - 60^{\circ}) = -\tan 60^{\circ} = -\sqrt{3} \). Also, \( \tan 45^{\circ} = 1 \).

Step 3: Calculate \( \tan(120^{\circ} + 45^{\circ}) \)

Substitute the values into the addition formula: \( \tan(120^{\circ} + 45^{\circ}) = \frac{-\sqrt{3} + 1}{1 + (-\sqrt{3})(1)} = \frac{-\sqrt{3} + 1}{1 - \sqrt{3}} \). Simplify this expression.

Step 4: Simplify the Addition Result

Multiply numerator and denominator by the conjugate \(1 + \sqrt{3}\) to rationalize: \( \frac{(-\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-3 + \sqrt{3} + \sqrt{3} + 3}{1 - 3} = \frac{2\sqrt{3}}{-2} = -\sqrt{3} \).

Step 5: Calculate \( \tan(120^{\circ} - 45^{\circ}) \)

Substitute the values into the subtraction formula: \( \tan(120^{\circ} - 45^{\circ}) = \frac{-\sqrt{3} - 1}{1 + (-\sqrt{3})(1)} = \frac{-\sqrt{3} - 1}{1 - \sqrt{3}} \). Simplify this expression.

Step 6: Simplify the Subtraction Result

Multiply numerator and denominator by the conjugate \(1 + \sqrt{3}\): \( \frac{(-\sqrt{3} - 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-3 - \sqrt{3} - \sqrt{3} - 1}{1 - 3} = \frac{-2 - 2\sqrt{3}}{-2} = 1 + \sqrt{3} \).

Key Concepts

Angle Sum and Difference

The concept of angle sum and difference is a fundamental tool in trigonometry. These are techniques used to find the trigonometric values of angles that are not typically found on the unit circle directly. Essentially, with these formulas, you can calculate the tangent, sine, and cosine of the sum or difference of two angles.
For tangent, the formulas are particularly useful:

• Sum Formula: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
• Difference Formula: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)

These formulas break down the calculations into easier chunks using simpler angles. With 120 degrees and 45 degrees, for example, one can explore these formulas to find the exact tangent.
What's great is these formulas don't just apply to tangent. Similar ones are applicable to sine and cosine, which makes them incredibly versatile for calculating complex angles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equation are defined. They are used to simplify trigonometric expressions and solve trigonometric equations.

• Identities like those involved with tangent have relations to triangles, giving them a geometrical meaning.
• For the tangent function, other important identities relate to complement and supplement angles, i.e., \( \tan(180^{\circ} - x) = -\tan(x) \).

In problems like the one given, these identities are crucial. They help you express complex angles in terms of their acute counterparts, leveraging known values such as \( \tan 45^{\circ} = 1 \).
These identities are tools that transform complicated trigonometric expressions into more manageable form.

Tangent Function

The tangent function, often represented as \( \tan \theta \), is a trigonometric function that relates an angle \( \theta \) in a right-angled triangle to the ratio of the opposite side to the adjacent side.
The tangent function has several important properties:

• Periodicity: \( \tan \theta \) repeats every 180 degrees, making it periodic.
• Undefined Points: Unlike sine and cosine, tangent is undefined for angles where cosine is zero (e.g., 90°, 270°) because it involves division by zero.
• Symmetry: The function is odd, meaning \( \tan(-\theta) = -\tan(\theta) \).

In calculations, such as the exercise, recognizing and leveraging these properties can significantly simplify the evaluation of tangent functions of various angles.
Utilizing properties like behavior near vertical asymptotes and periodicity can transform the approach to solving such problems with efficiency and accuracy.
In essence, the tangent function is not just about triangles but is integral in studying periodic phenomena.