Problem 60 - Precalculus Essentials

Question

Find the exact value of the following under the given conditions: (A) .\(\cos (\alpha+\beta)\) (B). \(\sin (\alpha+\beta)\) (C) \(\tan (\alpha+\beta)\) \(\tan \alpha=-\frac{4}{3}, \alpha\) lies in quadrant II, and \(\cos \beta=\frac{2}{3}, \beta\) lies in quadrant I.

Step-by-Step Solution

Answer

\(\cos (\alpha+\beta)=\frac{2}{15}(5-\sqrt{5})\), \(\sin (\alpha+\beta)=\frac{2}{15}(\sqrt{5}-2)\), \(\tan (\alpha+\beta)=\frac{\sqrt{5}-2}{5-\sqrt{5}}\)

Step 1: Determine the values of \(\sin \alpha\) and \(\cos \alpha\)

Given that \(\tan \alpha=-\frac{4}{3}\) and \(\alpha\) lies in quadrant II. We know that \(\tan =\frac{sin}{cos}\), so \(\sin \alpha=-\frac{4}{\sqrt{16+9}}\) and \(\cos \alpha=\frac{3}{\sqrt{16+9}}\) . The negative sign for \(\sin \alpha\) is because sine is negative in quadrant II.

Step 2: Determine the values of \(\sin \beta\) and \(\tan \beta\)

Given that \(\cos \beta=\frac{2}{3}\) and \(\beta\) lies in quadrant I. We can determine \(\sin \beta=\sqrt{1-\cos^2 \beta}=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3}\) . Also, \(\tan \beta = \frac{\sin \beta}{\cos \beta} = \frac{\sqrt{5}}{2}\) . The positive sign is because all ratios are positive in quadrant I.

Step 3: Calculate \(\cos (\alpha+\beta)\), \(\sin (\alpha+\beta)\) and \(\tan (\alpha+\beta)\)

Using the addition identities, we get \(\cos (\alpha+\beta)=\cos \alpha\cos \beta-\sin \alpha\sin \beta=\frac{3}{5}\times \frac{2}{3}- -\frac{4}{5}\times \frac{\sqrt{5}}{3}=\frac{2}{15}(5-\sqrt{5})\), \(\sin (\alpha+\beta)=\sin \alpha\cos \beta+\cos \alpha\sin \beta=-\frac{4}{5}\times \frac{2}{3}+\frac{3}{5}\times \frac{\sqrt{5}}{3}=\frac{2}{15}(\sqrt{5}-2)\), and \(\tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)}=\frac{\frac{2}{15}(\sqrt{5}-2)}{\frac{2}{15}(5-\sqrt{5})}=\frac{\sqrt{5}-2}{5-\sqrt{5}}\) by cross multiplying while simplifying.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the trigonometric functions that hold true for all values of the variables involved. They are essential tools for solving trigonometric equations, simplifying expressions, and evaluating angles. In this particular exercise, the angle addition formulas are crucial. These include:

• Cosine Addition: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• Sine Addition: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Tangent Addition: \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

These formulas allow us to find the exact trigonometric values of expressions involving the sum of two angles. By breaking down the problem using these identities, we can solve for specific values like \( \cos(\alpha + \beta) \), \( \sin(\alpha + \beta) \), and \( \tan(\alpha + \beta) \). Understanding and memorizing these identities can greatly simplify the task of working with trigonometric expressions.

Quadrant Analysis

Quadrant analysis helps us determine the sign and values of trigonometric functions based on the position of an angle in the coordinate system. The coordinate plane is divided into four quadrants:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, but cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, with sine and tangent being negative.

In this exercise, we're given that \( \alpha \) is located in Quadrant II and \( \beta \) in Quadrant I. This information determines the signs of the respective trigonometric functions. For \( \alpha \), since it's in Quadrant II:

• \( \sin \alpha \) is positive, but \( \cos \alpha \) is negative.

For \( \beta \), since it's in Quadrant I, all functions are positive. Knowing which quadrant an angle lies in helps you correctly apply signs in trigonometric identities.

Exact Trigonometric Values

Finding the exact trigonometric values involves more than just applying formulas; it requires finding precise numeric values typically without a calculator. In this problem, we discover the values by leveraging known identities and quadrant information:

• For \( \tan \alpha = -\frac{4}{3} \), we deduced \( \sin \alpha = \frac{-4}{5} \) and \( \cos \alpha = \frac{3}{5} \) because \( \alpha \) is in Quadrant II.
• For \( \cos \beta = \frac{2}{3} \), we use the Pythagorean identity \( \sin \beta = \sqrt{1 - \cos^2 \beta} = \frac{\sqrt{5}}{3} \), as \( \beta \) lies in Quadrant I.

These values are then plugged into the angle addition formulas to find the exact values of \( \cos(\alpha + \beta) \), \( \sin(\alpha + \beta) \), and \( \tan(\alpha + \beta) \). Performing these calculations precisely is key to obtaining correct results in problems involving angle addition.