Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

If \(\alpha\) and \(\beta\) are second-quadrant angles such that \(\sin \alpha=\frac{2}{3}\) and \(\cos \beta=-\frac{1}{3}\), find (a) \(\sin (\alpha+\beta)\) (b) \(\tan (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Step-by-Step Solution

Answer

(a) \(\sin(\alpha+\beta) = -\frac{2 + 2\sqrt{10}}{9}\), (b) \(\tan(\alpha+\beta)\) is complex to simplify; re-evaluate, (c) \(\alpha+\beta\) is in the third quadrant.

Step 1: Find cos(α) using Pythagorean Identity

Since \( \alpha \) is in the second quadrant where cosine is negative, use the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to find \( \cos \alpha \). Substitute \( \sin \alpha = \frac{2}{3} \):\[ \cos \alpha = - \sqrt{1 - \left(\frac{2}{3}\right)^2} = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3} \]

Step 2: Find sin(β) using Pythagorean Identity

Since \( \beta \) is in the second quadrant as well, where sine is positive, use the identity \( \sin^2 \beta + \cos^2 \beta = 1 \) to find \( \sin \beta \). Substitute \( \cos \beta = -\frac{1}{3} \):\[ \sin \beta = \sqrt{1 - \left(-\frac{1}{3}\right)^2} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} \]

Step 3: Apply Sine Angle Addition Formula

Calculate \( \sin(\alpha + \beta) \) using the angle addition formula:\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]Substitute \( \sin \alpha = \frac{2}{3}, \cos \alpha = -\frac{\sqrt{5}}{3}, \sin \beta = \frac{2\sqrt{2}}{3}, \cos \beta = -\frac{1}{3} \):\[ \sin(\alpha + \beta) = \frac{2}{3} \times \left(-\frac{1}{3}\right) + \left(-\frac{\sqrt{5}}{3}\right) \times \frac{2\sqrt{2}}{3} = -\frac{2}{9} - \frac{2\sqrt{10}}{9} = -\frac{2 + 2\sqrt{10}}{9} \]

Step 4: Apply Tangent Angle Addition Formula

Calculate \( \tan(\alpha + \beta) \) using the formula:\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]First, find \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = -\frac{2}{\sqrt{5}} \) and \( \tan \beta = \frac{\sin \beta}{\cos \beta} = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} \).Substitute:\[ \tan(\alpha + \beta) = \frac{-\frac{2}{\sqrt{5}} + (-2\sqrt{2})}{1 - \left(-\frac{2}{\sqrt{5}}\right)\left(-2\sqrt{2}\right)} = \frac{-\frac{2 + 2\sqrt{10}}{\sqrt{5}}}{1 - \frac{4\sqrt{2}}{\sqrt{5}}} \]

Step 5: Determine the Quadrant of (α + β)

Since both \( \sin(\alpha + \beta) \) and \( \cos(\alpha + \beta) \) are negative, \( \alpha + \beta \) is in the third quadrant. The negative sine indicates it lies in a quadrant where sine is negative, which are the third and fourth quadrants, but the negative cosine confirms it is the third quadrant.

Key Concepts

Second Quadrant Angles

Understanding angles in different quadrants is key to mastering trigonometry. In the Cartesian plane, angles are divided into four quadrants. The second quadrant is where angles are between 90° and 180°. Here, the sine function is positive, while cosine and tangent are negative. This is crucial because it dictates the sign of trigonometric ratios:

• Sine (\( heta\)) is positive.
• Cosine (\( heta\)) is negative.
• Tangent (\( heta\)) is negative.

This understanding helps students accurately determine trigonometric values and avoid simple sign errors when calculating angles and using identities related to specific quadrants.
Given a problem with angles \(\alpha\) and \(\beta\) in the second quadrant, expect \( an \alpha\) and \(\tan \beta\) to be negative, which directly impacts calculations to identify angles like \(\alpha + \beta\). Always verify the quadrant rules when your results deviate from expectations.

Sine and Cosine Identities

Sine and cosine identities are foundational in trigonometry. They help find missing information about angles or calculate more complex expressions. A key identity is the Pythagorean Identity:\[ \sin^2\theta + \cos^2\theta = 1 \]This aids in calculating unknown sine or cosine values if one is given. For angles in the second quadrant like \(\alpha\) and \(\beta\), the identities reflect the unique characteristics of the quadrant. If \(\sin \alpha = \frac{2}{3}\), to find \(\cos \alpha\), set up the equation:\[ \cos \alpha = -\sqrt{1 - \left(\frac{2}{3}\right)^2} \]The negative sign is crucial here due to the second quadrant’s cosine behavior.
These identities not only simplify complex problems but also act as checks for trigonometric operations and angle calculations. Always apply identities keeping in mind the specific quadrant characteristics, which will aid in avoiding wrong sign mistakes in the calculations.

Angle Addition Formulas

Angle addition formulas allow the calculation of trigonometric functions of two angles added together. They are:

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

For the given problem, utilizing the sine and cosine angle addition formulas helps calculate \(\sin(\alpha + \beta)\) and \(\tan(\alpha + \beta)\). Remember to substitute correctly, keeping signs consistent based on the quadrant. Angle addition formulas are powerful tools for expanding and simplifying complex trigonometric expressions, crucial when dealing with scenarios requiring the summation or difference of two angles.