Problem 11 - 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. $$ A=120^{\circ}, B=60^{\circ} $$

Step-by-Step Solution

Answer

\(\tan(180^\circ) = 0\) and \(\tan(60^\circ) = \sqrt{3}\).

Step 1: Recall the Formula for \(\tan(A + B)\)

The formula for \(\tan(A + B)\) is \(\frac{\tan A + \tan B}{1 - \tan A \tan B}\). We will use this formula to find \(\tan(120^\circ + 60^\circ)\).

Step 2: Find \(\tan 120^\circ\) and \(\tan 60^\circ\)

\(\tan 120^\circ = -\sqrt{3}\) because 120 degrees is in the second quadrant where tangent is negative and it corresponds to \(180^\circ - 60^\circ\). \(\tan 60^\circ = \sqrt{3}\).

Step 3: Substitute Values into \(\tan(A + B)\) Formula

Substitute \(\tan A = -\sqrt{3}\) and \(\tan B = \sqrt{3}\) into the formula: \(\frac{-\sqrt{3} + \sqrt{3}}{1 - (-\sqrt{3})(\sqrt{3})}\).

Step 4: Simplify \(\tan(A + B)\)

The numerator \(-\sqrt{3} + \sqrt{3} = 0\), and the denominator \(1 - (-3) = 4\). Hence, \(\tan(120^\circ + 60^\circ) = \frac{0}{4} = 0\).

Step 5: Recall the Formula for \(\tan(A - B)\)

The formula for \(\tan(A - B)\) is \(\frac{\tan A - \tan B}{1 + \tan A \tan B}\). We will use this to find \(\tan(120^\circ - 60^\circ)\).

Step 6: Substitute Values into \(\tan(A - B)\) Formula

Substitute \(\tan A = -\sqrt{3}\) and \(\tan B = \sqrt{3}\) into the formula: \(\frac{-\sqrt{3} - \sqrt{3}}{1 + (-\sqrt{3})(\sqrt{3})}\).

Step 7: Simplify \(\tan(A - B)\)

The numerator \(-\sqrt{3} - \sqrt{3} = -2\sqrt{3}\), and the denominator \(1 + (-3) = -2\). Simplifying, \(\tan(120^\circ - 60^\circ) = \frac{-2\sqrt{3}}{-2} = \sqrt{3}\).

Key Concepts

Tangent Addition Formula

The tangent addition formula is an essential identity in trigonometry. It helps calculate the tangent of the sum of two angles. The formula is given by: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] Using this formula, you can find the tangent of the sum of angles efficiently. Let's break it down to understand its application more clearly.

• First, find the tangent values for each angle separately.
• Substitute these values into the formula.
• Simplify the expression for an exact value.

In the example, for angles \(A = 120^{\circ}\) and \(B = 60^{\circ}\), the tangent values are \(-\sqrt{3}\) and \(\sqrt{3}\) respectively. Substituting these into our formula gives: \[ \tan(120^{\circ} + 60^{\circ}) = \frac{-\sqrt{3} + \sqrt{3}}{1 - (-\sqrt{3})(\sqrt{3})} \] Which simplifies to zero because the numerator becomes zero. This powerful formula makes working with tangent sums straightforward.

Tangent Subtraction Formula

Similar to the addition formula, the tangent subtraction formula allows you to find the tangent of the difference between two angles. The subtraction formula is: \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \] Using this formula requires a few easy steps:

• Evaluate the tangent of each angle.
• Insert these values into the formula.
• Simplify to reveal the tangent of the angle difference.

For the given angles \(A = 120^{\circ}\) and \(B = 60^{\circ}\), where \(\tan A = -\sqrt{3}\) and \(\tan B = \sqrt{3}\), the substitution looks like this: \[ \tan(120^{\circ} - 60^{\circ}) = \frac{-\sqrt{3} - \sqrt{3}}{1 + (-\sqrt{3})(\sqrt{3})} \] The computation simplifies to \(\sqrt{3}\), demonstrating the formula's utility in dealing with angle differences efficiently without confusion.

Simplifying Trigonometric Expressions

Simplification in trigonometry is more straightforward when you understand the properties of trigonometric functions. To simplify trigonometric expressions like \(\tan(A + B)\) or \(\tan(A - B)\), follow these techniques:

• Focus on the identities provided by formulas.
• Look at the signs and quadrants of the angles, as they affect the outcomes.
• Simplify each fraction by addressing like terms.

In our exercise, simplifying the formulas involved recognizing \(-\sqrt{3}\) and \(\sqrt{3}\) as the main components. The addition formula gave us zero due to a zero numerator, while the subtraction formula resolved complex terms into a simple \(\sqrt{3}\). Understanding the underlying identities and properties of tangent allows you to reduce seemingly complicated expressions into manageably simple results, aiding in precision and clarity when learning trigonometric solutions.