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Magazine Chapter 6: Problem 22
If \(\tan \alpha=-\frac{7}{24}\) and \(\cot \beta=\frac{3}{4}\) for a second- quadrant angle \(\alpha\) and a third-quadrant angle \(\boldsymbol{\beta},\) find (a) \(\sin (\alpha+\beta)\) \(\langle b\rangle \cos (\alpha+\beta)\) \(\langle c) \tan (\alpha+\beta)\) (d) \(\sin (\alpha-\beta)\) (e) \(\cos (\alpha-\beta)\) (f) \(\tan (\alpha-\beta)\)
Short Answer
Expert verified
Calculate trigonometric identities using given angles' characteristics.
Step by step solution
01
Determine sine and cosine for \(\alpha\)
Given that \(\tan \alpha = -\frac{7}{24}\) and \(\alpha\) is in the second quadrant, use the identity \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\). In the second quadrant, \(\sin \alpha > 0\) and \(\cos \alpha < 0\). Assume \(\cos \alpha = -24x\) and \(\sin \alpha = 7x\), resulting in \(\frac{7x}{-24x} = -\frac{7}{24}\). Hence, \(x^2 = \frac{1}{} (24^2 + 7^2)\). Solve for \(x\) and determine \(\sin \alpha\) and \(\cos \alpha\).
02
Determine sine and cosine for \(\beta\)
Given that \(\cot \beta = \frac{3}{4}\) and \(\beta\) is in the third quadrant, use the identity \(\cot \beta = \frac{\cos \beta}{\sin \beta}\). In the third quadrant, both \(\sin \beta\) and \(\cos \beta\) are negative. Assume \(\cos \beta = -3y\) and \(\sin \beta = -4y\), resulting in \(\frac{-3y}{-4y} = \frac{3}{4}\). The hypotenuse \(r = \sqrt{3^2 + 4^2} = 5\), so \(\cos \beta = -\frac{3}{5}\) and \(\sin \beta = -\frac{4}{5}\).
03
Calculate \(\sin (\alpha + \beta)\)
Use the identity \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\). Substitute the values: \(\sin \alpha = \frac{7}{25}\), \(\cos \alpha = -\frac{24}{25}\), \(\cos \beta = -\frac{3}{5}\), and \(\sin \beta = -\frac{4}{5}\). Calculate the result for \(\sin (\alpha + \beta)\).
04
Calculate \(\cos (\alpha + \beta)\)
Use the identity \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\). Substitute the known values and compute \(\cos (\alpha + \beta)\).
05
Calculate \(\tan (\alpha + \beta)\)
Use the identity \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\). Identify \(\tan \beta = \frac{4}{3}\), and substitute into the expression. Simplify to find \(\tan (\alpha + \beta)\).
06
Calculate \(\sin (\alpha - \beta)\)
Use the identity \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\). Substitute the known values and calculate \(\sin (\alpha - \beta)\).
07
Calculate \(\cos (\alpha - \beta)\)
Use the identity \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\). Substitute the known values and compute \(\cos (\alpha - \beta)\).
08
Calculate \(\tan (\alpha - \beta)\)
Use the identity \(\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\). Substitute \(\tan \alpha = -\frac{7}{24}\) and \(\tan \beta = \frac{4}{3}\) into the expression. Simplify to find \(\tan (\alpha - \beta)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle Addition Identity
The Angle Addition Identity is a fundamental concept in trigonometry that helps us find the sine, cosine, and tangent of the sum of two angles. When dealing with trigonometric expressions like \(\sin(\alpha + \beta)\), \(\cos(\alpha + \beta)\), or \(\tan(\alpha + \beta)\), this identity comes into play. Here's what you need to remember:
A big hint here is to carefully identify the known values of \(\sin\), \(\cos\), and \(\tan\) for each angle before plugging them into these formulas.
- For sine: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).
- For cosine: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).
- For tangent: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).
A big hint here is to carefully identify the known values of \(\sin\), \(\cos\), and \(\tan\) for each angle before plugging them into these formulas.
Angle Subtraction Identity
The Angle Subtraction Identity is similar to the Angle Addition Identity but is used for finding trigonometric values when subtracting angles. It is just as useful and often required alongside its addition counterpart.
- For sine: \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\).
- For cosine: \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).
- For tangent: \(\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\).
Trigonometric Functions
Trigonometric functions form the backbone of trigonometry and are essential in solving problems involving angles. The key functions include sine, cosine, and tangent, and each has a unique ratio derived from a right-angled triangle or on the unit circle. Here's a quick look at these functions:
- \(\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}\)
- \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
- \(\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{\sin \theta}{\cos \theta}\)
Quadrants of the Unit Circle
The unit circle is a crucial tool in trigonometry that helps determine the trigonometric values of angles, particularly their sign. It is divided into four quadrants, each with distinct characteristics based on the signs of sine and cosine values of angles:
- First Quadrant: Both sine and cosine are positive.
- Second Quadrant: Sine is positive, but cosine is negative.
- Third Quadrant: Both sine and cosine are negative.
- Fourth Quadrant: Sine is negative, whereas cosine is positive.
