Problem 15 - 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. $$ A=\pi, B=\frac{\pi}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the tangent addition and subtraction formulas

The tangent addition formula is given by \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) and the tangent subtraction formula is \( \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \). We will use these formulas to find \( \tan(A+B) \) and \( \tan(A-B) \).

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

For \( A = \pi \), we know that \( \tan \pi = 0 \) because \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sin \pi = 0 \) while \( \cos \pi = -1 \), so \( \tan \pi = 0 \). \( B = \frac{\pi}{3} \), and \( \tan \frac{\pi}{3} = \sqrt{3} \) as it is a standard angle.

Step 3: Calculate \( \tan(A+B) \) using the addition formula

Substitute \( \tan A = 0 \) and \( \tan B = \sqrt{3} \) into the formula: \( \tan(A+B) = \frac{0 + \sqrt{3}}{1 - 0 \cdot \sqrt{3}} = \frac{\sqrt{3}}{1} = \sqrt{3} \).

Step 4: Calculate \( \tan(A-B) \) using the subtraction formula

Substitute \( \tan A = 0 \) and \( \tan B = \sqrt{3} \) into the formula: \( \tan(A-B) = \frac{0 - \sqrt{3}}{1 + 0 \cdot \sqrt{3}} = \frac{-\sqrt{3}}{1} = -\sqrt{3} \).

Key Concepts

Tangent Function

The tangent function, an important element in trigonometry, measures the ratio of the lengths of two sides in a right triangle. Defined as the ratio of the opposite to the adjacent side with respect to an angle, the formula is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This means it is the sine of the angle divided by the cosine of the angle.

Understanding the tangent function is crucial for solving problems involving angles and triangles, particularly in trigonometric equations, which often include addition and subtraction formulas. These formulas allow us to find the tangent of sums and differences of angles, making it simpler to handle complex trigonometric expressions.

When using the tangent addition formula \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) and the subtraction formula \( \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \), keep in mind that they help decompose or simplify the tangent of more complex angles by breaking them down into known values.

Standard Angles

Standard angles, such as \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and their multiples are fundamental in trigonometry. These angles are frequently used because they have easily recognizable trigonometric values.

For instance:

• \( \tan 0 = 0 \)
• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)
• \( \tan \frac{\pi}{4} = 1 \)
• \( \tan \frac{\pi}{3} = \sqrt{3} \)
• \( \tan \frac{\pi}{2} \) is undefined because cosine is zero at this angle.

When solving trigonometric problems, recognizing the tangent values for these standard angles can significantly simplify calculations. This was evident in our exercise where \( \tan \pi = 0 \) and \( \tan \frac{\pi}{3} = \sqrt{3} \) were used to determine \( \tan(A+B) \) and \( \tan(A-B) \).

Familiarity with these angles and their properties is key in simplifying and solving trigonometric equations and identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable they include. These are valuable tools in simplifying and solving trigonometric expressions. For the tangent function, the addition and subtraction identities, mentioned earlier, are pivotal.

In general:

• Trigonometric identities are used to prove other equations.
• They simplify expressions, making them easier to evaluate or integrate.
• Identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( 1 + \tan^2 \theta = \sec^2 \theta \) are foundational.

Using identities ensures precision across mathematical calculations and is essential for solving complex trigonometric equations, whether you're breaking down angles or proving the equality of different expressions.

By deeply understanding identities and their applications, we can handle various trigonometric challenges efficiently, as illustrated by the steps in our solution journey.