Problem 3 - 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. $$ A=45^{\circ}, B=30^{\circ} $$

Step-by-Step Solution

Answer

\(\tan(A + B) = 2 + \sqrt{3}\) and \(\tan(A - B) = 2 - \sqrt{3}\).

Step 1: Recall the Formulas

The tangent of a sum of two angles is given by the formula \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \). Similarly, the tangent of the difference of two angles is given by \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B} \). We'll use these formulas to find \(\tan(A+B)\) and \(\tan(A-B)\).

Step 2: Calculate \(\tan A\) and \(\tan B\)

Given \( A = 45^{\circ} \) and \( B = 30^{\circ} \), we know that \( \tan 45^{\circ} = 1 \) and \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \). We'll use these values in our formulas.

Step 3: Apply the Sum Formula

Substitute \( \tan A = 1 \) and \( \tan B = \frac{1}{\sqrt{3}} \) into the sum formula: \[ \tan(A + B) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \] Simplifying gives: \[ \tan(A + B) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \] Let's simplify further.

Step 4: Simplify the Expression for \(\tan(A + B)\)

To simplify the expression \( \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \), multiply the numerator and denominator by \( \sqrt{3} \) to get rid of the fractions: \[ \frac{(1 + \frac{1}{\sqrt{3}}) \cdot \sqrt{3}}{(1 - \frac{1}{\sqrt{3}}) \cdot \sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \] Now use the difference of squares to simplify further: \[ \tan(A + B) = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 + 2\sqrt{3} + 1}{2} = 2 + \sqrt{3} \].

Step 5: Apply the Difference Formula

Substitute \( \tan A = 1 \) and \( \tan B = \frac{1}{\sqrt{3}} \) into the difference formula: \[ \tan(A - B) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} \] Simplifying gives: \[ \tan(A - B) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} \] Proceed with further simplification.

Step 6: Simplify the Expression for \(\tan(A - B)\)

To simply \( \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} \), multiply the numerator and the denominator by \( \sqrt{3} \): \[ \frac{(1 - \frac{1}{\sqrt{3}}) \cdot \sqrt{3}}{(1 + \frac{1}{\sqrt{3}}) \cdot \sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \] The conjugate of the denominator can be used to simply further: \[ \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 - 2\sqrt{3} + 1}{2} = 2 - \sqrt{3} \].

Key Concepts

Tangent Subtraction Formula

To understand the tangent subtraction formula, it's essential to recognize it as part of trigonometric identities that help simplify the calculation of trigonometric functions involving angle sums or differences. The formula is: \[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B} \]This formula allows you to find the tangent of the difference between two angles, which can be very useful in geometry and trigonometry.

• The subtraction formula is derived from the sine and cosine subtraction identities.
• Using this formula provides a simplified way to calculate the tangent of a difference without converting everything into base trigonometric functions like sine or cosine.
• When applied, it can reduce computational complexity and help solve problems more efficiently.

It is handy, especially when dealing with standard angle measures or when applying it to solve problems with known tangent values, such as in the exercise where angles 45° and 30° were used.

Angle Sum Identity

The angle sum identity is a critical trigonometric concept involving finding the trigonometric function of the sum of two angles. In the context of tangent, the angle sum identity is expressed as:\[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \]This formula provides an effective way to find the tangent of the sum of two angles. Here's how it can help:

• It calculates the tangent of an angle by summing the tangents of two separate angles.
• The identity is derived from the foundational trigonometric identities for sine and cosine.
• This makes solving complex trigonometric equations simpler, without manually finding sine and cosine values.

By using this identity, problems like finding \( \tan(45^\circ + 30^\circ) \) become straightforward, changing a potentially complex task into a simple application of the formula.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. They are fundamental in simplifying complex trigonometric expressions and solving related equations.

• These identities are derived from the basic sine, cosine, and tangent functions, often involving their relationships and transformations.
• Some common identities include Pythagorean identities, angle addition and subtraction identities, and double angle identities.
• They serve as tools to transform complicated trigonometric expressions into simpler forms or solve equations, making them extremely powerful mathematical tools.

Understanding these identities allows one to manipulate and solve equations efficiently, as illustrated when finding \(\tan(A+B)\) and \(\tan(A-B)\) using the respective formulas. They lay the groundwork for more advanced mathematical studies in trigonometry and calculus.