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Chapter 5: Problem 50

Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\cot (v-u)$$

Short Answer

Expert verified
\(\cot(v - u) = \frac{117}{44}\)

Step by step solution

01

- Find \(\sin u\) and \(\cos u\) with given information

It's given that \(\sin u = -\frac{7}{25}\). In order to find \(\cos u\), we can use the Pythagorean trigonometric identity \(\sin^2u + \cos^2u = 1\). Solving for \(\cos u\), we obtain \(\cos u = \sqrt{1 - \sin^2u} = \sqrt{1 - (-\frac{7}{25})^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}}\). Since \(u\) is in Quadrant III, \(\cos u\) is negative, so \(\cos u = -\frac{24}{25}\).
02

- Find \(\cos v\) with given information

It's given that \(\cos v = -\frac{4}{5}\). We can find \(\sin v\) using the Pythagorean trigonometric identity similar to previous step. Therefore, \(\sin v = \sqrt{1 - (-\frac{4}{5})^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}}\). As \(v\) is in Quadrant III, \(\sin v\) is negative, so \(\sin v = -\frac{3}{5}\).
03

- Computing \(\sin(u-v)\) and \(\cos(u-v)\)

Now we can find \(\sin(u-v)\) using the formula: \(\sin(u-v) = \sin u \cos v - \cos u \sin v = (-\frac{7}{25})(-\frac{4}{5}) - (-\frac{24}{25})(-\frac{3}{5}) = \frac{28}{125} - \frac{72}{125} = -\frac{44}{125}\). Similarly, we compute \(\cos(u-v)\) using its formula: \(\cos(u-v) = \cos u \cos v + \sin u \sin v = (-\frac{24}{25})(-\frac{4}{5}) + (-\frac{7}{25})(-\frac{3}{5}) = \frac{96}{125} + \frac{21}{125} = \frac{117}{125}\).
04

- Compute \(\cot(v-u)\)

Now we can find \(\cot(v-u)\). The formula for cotangent in terms of sine and cosine is \(\cot x = \frac{\cos x}{\sin x}\). Thus, \(\cot(v - u) = - \cot(u - v)\) since \(\cot\) is an odd function, and \(cot(-x) = - cot(x)\). Therefore, \(\cot(v - u) = -\frac{\cos(u - v)}{\sin(u - v)} = -\frac{\frac{117}{125}}{-\frac{44}{125}} = \frac{117}{44}\).

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