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Chapter 6: Problem 26

If \(\tan \alpha=-\frac{7}{24}\) and \(\cot \beta=\frac{3}{4}\) for a second- quadrant angle \(\alpha\) and a third-quadrant angle \(\beta,\) find (a) \(\sin (\alpha+\beta)\) (b) \(\cos (\alpha+\beta)\) (c) \(\tan (\alpha+\beta)\) (d) \(\sin (\alpha-\beta)\) \((e) \cos (\alpha-\beta)\) (f) \(\tan (\alpha-\beta)\)

Short Answer

Expert verified
(a) \(\frac{3}{5}\), (b) \(\frac{4}{5}\), (c) \(\frac{3}{4}\), (d) \(-\frac{117}{125}\), (e) \(\frac{44}{125}\), (f) \(-\frac{117}{44}\)

Step by step solution

01

Use given values to express sine and cosine of 伪 and 尾

Given that \( \tan \alpha = -\frac{7}{24} \) in the second quadrant, we have \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \). Since \( \tan \alpha \) is negative in the second quadrant and \( \sin \alpha \) is positive, \( \cos \alpha \) must be negative. Using the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \), we find \( \sin \alpha = \frac{7}{\sqrt{24^2 + 7^2}} = \frac{7}{25} \) and \( \cos \alpha = -\frac{24}{25} \).Given \( \cot \beta = \frac{3}{4} \) in the third quadrant, \( \cot \beta = \frac{\cos \beta}{\sin \beta} \). Since both \( \sin \beta \) and \( \cos \beta \) are negative in the third quadrant, we find \( \cos \beta = -\frac{3}{\sqrt{3^2 + 4^2}} = -\frac{3}{5} \) and \( \sin \beta = -\frac{4}{5} \).
02

Calculate sin(伪 + 尾) using sum formula

Use the sine addition formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \). Substituting the values:\[ \sin(\alpha + \beta) = \left(\frac{7}{25}\right)\left(-\frac{3}{5}\right) + \left(-\frac{24}{25}\right)\left(-\frac{4}{5}\right) \]Simplifying:\[ \sin(\alpha + \beta) = -\frac{21}{125} + \frac{96}{125} = \frac{75}{125} = \frac{3}{5} \]
03

Calculate cos(伪 + 尾) using sum formula

Use the cosine addition formula: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \). Substitute the values:\[ \cos(\alpha + \beta) = \left(-\frac{24}{25}\right)\left(-\frac{3}{5}\right) - \left(\frac{7}{25}\right)\left(-\frac{4}{5}\right) \]Simplifying:\[ \cos(\alpha + \beta) = \frac{72}{125} + \frac{28}{125} = \frac{100}{125} = \frac{4}{5} \]
04

Calculate tan(伪 + 尾) using sine and cosine results

The tangent of a sum is \( \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \). Using the previously found values:\[ \tan(\alpha + \beta) = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]
05

Calculate sin(伪 - 尾) using difference formula

Use the sine difference formula: \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \). Substitute the values:\[ \sin(\alpha - \beta) = \left(\frac{7}{25}\right)\left(-\frac{3}{5}\right) - \left(-\frac{24}{25}\right)\left(-\frac{4}{5}\right) \]Simplifying:\[ \sin(\alpha - \beta) = -\frac{21}{125} - \frac{96}{125} = -\frac{117}{125} \]
06

Calculate cos(伪 - 尾) using difference formula

Use the cosine difference formula: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \). Substitute the values:\[ \cos(\alpha - \beta) = \left(-\frac{24}{25}\right)\left(-\frac{3}{5}\right) + \left(\frac{7}{25}\right)\left(-\frac{4}{5}\right) \]Simplifying:\[ \cos(\alpha - \beta) = \frac{72}{125} - \frac{28}{125} = \frac{44}{125} \]
07

Calculate tan(伪 - 尾) using sine and cosine results

The tangent of a difference is \( \tan(\alpha - \beta) = \frac{\sin(\alpha - \beta)}{\cos(\alpha - \beta)} \). Using the previously found values:\[ \tan(\alpha - \beta) = \frac{-\frac{117}{125}}{\frac{44}{125}} = -\frac{117}{44} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum and Difference Formulas
To tackle the problem, you need to understand the sum and difference formulas in trigonometry. These formulas help in finding the sine, cosine, and tangent of angles that are the result of adding or subtracting two other angles. The sum formulas are:
  • 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: \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)
Similarly, the difference formulas are:
  • 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: \( \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \)
These formulas are essential when you know the values of sine, cosine, or tangent for individual angles and need to find them for their sum or difference.
Second Quadrant Angles
Understanding the behavior of angles in different quadrants is critical. In the second quadrant, angles range from 90掳 to 180掳, or \( \frac{\pi}{2} \) to \( \pi \) radians. The important things to note about second quadrant angles are:
  • \( \sin \theta \) is positive.
  • \( \cos \theta \) is negative.
  • \( \tan \theta \) is negative.
Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), its negativity implies that sine and cosine have opposite signs, which matches with the second quadrant behaviors: positive sine and negative cosine. This context is especially useful when converting the information about \( \tan \alpha \) to \( \sin \alpha \) and \( \cos \alpha \) in the given problem.
Third Quadrant Angles
In the third quadrant, angles fall between 180掳 and 270掳, or \( \pi \) to \( \frac{3\pi}{2} \) radians. For third quadrant angles:
  • Both \( \sin \theta \) and \( \cos \theta \) are negative.
  • \( \tan \theta \) is positive because it is the ratio of two negative numbers.
Knowing that both sine and cosine are negative in this quadrant helps when you need to calculate each function's exact value, as both will draw from the negativity of the quadrant. It is particularly relevant in our context since angle \( \beta \) lies in this quadrant.
Sine Function
The sine function is one of the primary trigonometric functions that describes the y-coordinate of a point on the unit circle. For any angle \( \theta \), \( \sin \theta \) can be defined as:
  • The opposite side over the hypotenuse in a right triangle.
  • The y-coordinate of the corresponding point on the unit circle.
In this exercise, recognizing that \( \sin \alpha \) is positive in the second quadrant and \( \sin \beta \) is negative in the third quadrant helps determine the values needed when using the sum and difference formulas for sine.
Cosine Function
Cosine relates to the x-coordinate on the unit circle for a given angle. It is defined as:
  • The adjacent side over the hypotenuse in a right triangle.
  • The x-coordinate of the corresponding point on the unit circle.
For second quadrant angles like \( \alpha \), \( \cos \alpha \) is negative, reflecting the x-position being left of the y-axis. Similarly, \( \cos \beta \) is negative in the third quadrant for \( \beta \), showing a negative x-component. This negativity must be considered in calculations involving cosine for sum and difference scenarios.
Tangent Function
The tangent function is a ratio of sine over cosine, that is, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The signs of sine and cosine influence the sign of tangent:
  • Tangent is positive when sine and cosine share the same sign.
  • Tangent is negative when sine and cosine have different signs.
In our example, \( \tan \alpha \) is negative in the second quadrant, which matches because sine and cosine have opposite signs. However, \( \tan \beta \) is positive in the third quadrant, as both sine and cosine are negative, resulting in a positive ratio. Understanding these relations aids in the application of tangent sum and difference formulas.

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Most popular questions from this chapter

Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0$$. Simplify the resulting expression. $$\frac{x^{4}}{\left(a^{2}+x^{2}\right)^{2}}$$

Use the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an identity. Verify this identity. $$f(x)=\frac{\sin x-\sin ^{3} x}{\cos ^{4} x+\cos ^{2} x \sin ^{2} x}$$

Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0$$ Simplify the resulting expression. $$\frac{\left(x^{2}-a^{2}\right)^{2}}{x^{2}}$$

Express in terms of the cosine function with exponent 1. $$\sin ^{4} 2 x$$

Estimate the solutions of the equation in the interval \([-\pi, \pi]\). $$\ln \left(1+\sin ^{2} x\right)=\cos x$$
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