Problem 16 - 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. $$ A=\frac{5 \pi}{6}, B=\frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

\(\tan(A+B) = -\sqrt{3}\) and \(\tan(A-B) = 0\).

Step 1: Recall and Use the Tangent Addition and Subtraction Formulas

To find \( \tan(A + B) \) and \( \tan(A - B) \), we use the formulas:\[ \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \]and\[ \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \]First, calculate \( \tan(A) \) and \( \tan(B) \).

Step 2: Determine \(\tan(A)\) and \(\tan(B)\) for Given Angles

\( A = \frac{5 \pi}{6} \) and \( B = \frac{5 \pi}{6} \). The tangent of an angle \( \frac{5\pi}{6} \) can be calculated using the identity:\[ \tan\left(\frac{5 \pi}{6}\right) = \tan(\pi - \frac{\pi}{6}) \]The tangent of an angle in the form \( \pi - \theta \) is equal to the negative of \( \tan(\theta) \). Therefore:\[ \tan\left(\frac{5 \pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}} \]So, \( \tan(A) = -\frac{1}{\sqrt{3}} \) and \( \tan(B) = -\frac{1}{\sqrt{3}} \).

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

Substitute the values for \( \tan(A) \) and \( \tan(B) \) into the \( \tan(A + B) \) formula:\[ \tan(A + B) = \frac{-\frac{1}{\sqrt{3}} + -\frac{1}{\sqrt{3}}}{1 - \left(-\frac{1}{\sqrt{3}}\right)\left(-\frac{1}{\sqrt{3}}\right)} \]Simplifying that expression:\[ \tan(A + B) = \frac{-\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{-\frac{2}{\sqrt{3}}}{\frac{2}{3}} = -\sqrt{3} \]

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

Since \( A = B \), then \( A - B = 0 \). The tangent of 0 is 0:\[ \tan(A - B) = \tan(0) = 0 \]

Step 5: Summarize the Results

Thus, the exact values that we have found are:\( \tan(A + B) = -\sqrt{3} \)\( \tan(A - B) = 0 \).

Key Concepts

Tangent Identity

The tangent identity is an essential concept in trigonometry that helps us understand the behavior and relationships of angles through tangent functions. Specifically, the formulas for tangent addition and subtraction are vital tools to calculate the tangent of a sum or difference of two angles. These identities are given by the expressions:

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

The use of these formulas allows us to derive other values from known tangents, which can be crucial in solving more complex trigonometric problems. Becoming familiar with these identities can vastly simplify calculations and enhance your understanding of the tangent function's properties.

Angle Addition

Angle addition is a fundamental concept in trigonometry that involves summing angles and determining trigonometric function values of the result. This is particularly useful when you need to find trigonometric values of non-standard angles. By using the angle addition formula for tangent, we can solve problems that involve the angles' sum or their difference.In our exercise example, we have angles

• \( A = \frac{5 \pi}{6} \)
• \( B = \frac{5 \pi}{6} \)

Using these, we calculate

• \( \tan(A + B) \)
• \( \tan(A - B) \)

through the application of the tangent addition formula. Understanding angle addition helps us tackle problems where direct evaluation isn't straightforward due to the angles involved, making it an invaluable tool in trigonometric calculations.

Trigonometric Formulas

Trigonometric formulas encompass a wide variety of equations important for managing relationships between angles and sides of right triangles. Among these, formulas for tangent, sine, and cosine are crucial. For tangent specifically, we use the tangent addition and subtraction formulas to handle compound angles. When angles are given like

• \( A = \frac{5 \pi}{6} \)
• \( B = \frac{5 \pi}{6} \)

the tangent formulas allow us to address their combination or difference by converting it into manageable computations using known tangent identities. This supports understanding complex angles by relating them back to more straightforward and known angles, demonstrating the importance of trigonometric formulas in both theoretical and applied mathematics.

Exact Trigonometric Values

Exact trigonometric values refer to specific known values for the angles commonly encountered in trigonometric problems, such as 30°, 45°, 60°, or their radian equivalents. These values are derived from fundamental trigonometric identities or specialized triangles like the 30-60-90 triangle.For example, the tangent of 30° (or \( \frac{\pi}{6} \)) is known to be

• \( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \)

In our exercise, the angle

• \( A = \frac{5 \pi}{6} \) can be understood through its relation to \( \pi - \frac{\pi}{6} \)

Using the identity for tangent reversals, we find that

• \( \tan(\frac{5 \pi}{6}) = -\frac{1}{\sqrt{3}} \)

Applying known exact values allows for solving complex problems with precision and ease, ensuring that the trigonometric functions compute within elementary properties for accurate results.