Problem 7 - 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. $$ A=180^{\circ}, B=45^{\circ} $$

Step-by-Step Solution

Answer

\(\tan(225^{\circ}) = 1\), \(\tan(135^{\circ}) = -1\).

Step 1: Recognize the Tangent Addition and Subtraction Formulas

We want to find \( \tan(A+B) \) and \( \tan(A-B) \). The formulas for these are:\[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]\[ \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \].

Step 2: Calculate Tangent Values for A and B

Now, let's calculate the tangent values for \(A = 180^{\circ}\) and \(B = 45^{\circ}\):- \(\tan(180^{\circ}) = 0\) because \(180^{\circ}\) is equivalent to \(\pi\) radians, where the tangent is zero.- \(\tan(45^{\circ}) = 1\) because in a 45-degree angle, the opposite and adjacent sides are equal.

Step 3: Compute \(\tan(A+B)\) Using the Addition Formula

Substitute the values into the tangent addition formula:\[ \tan(180^{\circ} + 45^{\circ}) = \frac{\tan 180^{\circ} + \tan 45^{\circ}}{1 - \tan 180^{\circ} \times \tan 45^{\circ}} = \frac{0 + 1}{1 - 0 \times 1} = 1 \].

Step 4: Compute \(\tan(A-B)\) Using the Subtraction Formula

Substitute the values into the tangent subtraction formula:\[ \tan(180^{\circ} - 45^{\circ}) = \frac{\tan 180^{\circ} - \tan 45^{\circ}}{1 + \tan 180^{\circ} \times \tan 45^{\circ}} = \frac{0 - 1}{1 + 0 \times 1} = -1 \].

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that connect the angles and sides of a triangle. They enable us to establish relationships between various trigonometric functions like sine, cosine, and tangent. These identities are incredibly useful in transforming complex trigonometric expressions into simpler forms, making calculations much easier.

One of the most commonly used trigonometric identities is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

Other important identities include the reciprocal identities, which express one trigonometric function in terms of another, such as:\( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Understanding these identities is crucial as they form the building blocks for more complicated equations, like the angle addition and subtraction formulas discussed in this exercise.

Angle Addition Formula

The angle addition formula is a crucial tool for calculating the sine, cosine, or tangent of the sum of two angles. For this exercise, we primarily focus on the tangent addition formula. It allows us to find \( \tan(A+B) \) using known values of \( \tan A \) and \( \tan B \).

The tangent addition formula is given by:

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

The denominator of this formula ensures that the tangent does not become undefined by accounting for the potential cases where the product \( \tan A \tan B = 1 \).

This formula is really helpful when dealing with angles where direct calculation of the tangent function would be challenging. In the example of \( A = 180^{\circ} \) and \( B = 45^{\circ} \), the formula simplifies the work because the tangent of these angles individually are easy to determine: \( \tan(180^{\circ}) = 0 \) and \( \tan(45^{\circ}) = 1 \).

By substituting these known values into the formula, we discover that \( \tan(225^{\circ}) = 1 \), which highlights the efficiency and practicality of using the angle addition formula.

Angle Subtraction Formula

The angle subtraction formula is similar in form and purpose to the addition formula, but it is used to find the trigonometric function of the difference between two angles. In this exercise, the focus is on \( \tan(A-B) \).

The formula for the tangent subtraction is:

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

Just like the angle addition formula, it uses known tangent values of specific angles to calculate the tangent of their difference efficiently.

Using \( A = 180^{\circ} \) and \( B = 45^{\circ} \), we find \( \tan(135^{\circ}) \). Given that \( \tan 180^{\circ} = 0 \) and \( \tan 45^{\circ} = 1 \), substituting these into the subtraction formula:

• \( \tan(135^{\circ}) = \frac{0 - 1}{1 + 0 \times 1} = -1 \)

Hence, angle subtraction formulas streamline complex calculations and provide exact values in a straightforward manner.