Problem 8 - 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. $$ A=180^{\circ}, B=60^{\circ} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Tangent Addition and Subtraction Formulas

To solve for \( \tan(A+B) \) and \( \tan(A-B) \), we utilize the tangent addition formula \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) and the subtraction formula \( \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \).

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

For \( A = 180^\circ \), \( \tan(180^\circ) = 0 \) because tangent is sine over cosine, and at \( 180^\circ \), sine is 0 and cosine is -1. For \( B = 60^\circ \), \( \tan(60^\circ) = \sqrt{3} \).

Step 3: Calculate \( \tan(A+B) \)

Substitute \( \tan A = 0 \) and \( \tan B = \sqrt{3} \) into the addition formula: \[ \tan(180^\circ + 60^\circ) = \frac{0 + \sqrt{3}}{1 - 0 \cdot \sqrt{3}} = \sqrt{3} \].

Step 4: Calculate \( \tan(A-B) \)

Substitute \( \tan A = 0 \) and \( \tan B = \sqrt{3} \) into the subtraction formula: \[ \tan(180^\circ - 60^\circ) = \frac{0 - \sqrt{3}}{1 + 0 \cdot \sqrt{3}} = -\sqrt{3} \].

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables. They serve as essential tools for simplifying and solving trigonometric expressions.

These identities include fundamental relationships like the Pythagorean identities:

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

Other important identities help us connect different trigonometric functions, such as the reciprocal identities:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Having a strong grasp of these identities allows us not only to solve equations but also to transform expressions into more manageable forms. Whether you are a student or someone tackling trigonometry in a practical application, recognizing and applying these identities can significantly ease your problem-solving process.

Angle Sum and Difference Identities

The angle sum and difference identities are crucial for evaluating trigonometric functions at various angles. Specifically for tangent, the sum formula is:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] and the difference formula is:

\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \] These identities help us break down complex angles into simpler components. They are particularly useful for angles not commonly found on the unit circle, such as 105° or 135°, by allowing these angles to be expressed as sums or differences of angles with known trigonometric values.

For instance, knowing the values of tangent at 180° and 60°, we can easily compute the tangent of 240° or 120° using the formulas above. This method is not only critical in theoretical contexts but also in engineering and physics problems where angle calculations are frequent.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the primary functions in trigonometry, depicted as the ratio of the sine and cosine functions:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

This function is periodic with a period of 180° or \( \pi \) radians, meaning it repeats its values every 180° as it goes through its cycle. Its graph is an endless series of repeating curves that feature asymptotes at odd multiples of 90°, where the tangent function is undefined.

The tangent function is essential in many trigonometric calculations because it provides a straightforward relationship between angles and side ratios in right triangles. It is particularly helpful when needing to assess slopes, angles of inclination, and distance measurements in real-world scenarios. Understanding tangent's behavior and its key values, like \( \tan(45°) = 1 \) or \( \tan(60°) = \sqrt{3} \), is vital for solving problems efficiently.

Trigonometry Problem Solving

Engaging in trigonometry problem solving involves strategically employing trigonometric identities and functions to find unknown values or prove certain properties.

The problem we tackled required finding \( \tan(A+B) \) and \( \tan(A-B) \). Here's how you might approach such problems:

• **Step 1**: Understand the given values and what you need to find. Recognize the need to use specific identities, such as tangent addition and subtraction formulas in this case.
• **Step 2**: Calculate the trigonometric values of given angles, like \( \tan A \) and \( \tan B \). This requires knowledge of standard angle values and their properties.
• **Step 3**: Substitute these values into the appropriate identity. Simplify the expression to find the solution.

By following these steps, we reduce errors and clarify the path to the solution. Such structured problem solving is beneficial beyond mathematics, sharpening one's logical reasoning and analytical skills in complex problem scenarios.