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Chapter 5: Problem 49

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.) $$\tan (u-v)$$

Short Answer

Expert verified
The exact value of \(\tan (u-v)\) is \(\frac{56}{105}\).

Step by step solution

01

Find the Cosine and Sine Values

Since \(u\) and \(v\) are both in Quadrant III, both sine and cosine values are negative. We can use Pythagorean identity \(\sin^2u + \cos^2u = 1\), to find \(cos u\). Rearranging the equation, we get \(cos^2u = 1 - \sin^2u\). Substituting \(\sin u = -\frac{7}{25}\), we get \(cos u = -\sqrt{1-(-\frac{7}{25})^2}\) which equals \(-\frac{24}{25}\). Similarly, for \(v\), we use the equation \(\cos^2v + \sin^2v = 1\) to compute \(sin v = -\sqrt{1-(-\frac{4}{5})^2}\) which equals \(-\frac{3}{5}\). So, \(\cos u = -\frac{24}{25}\) and \(\sin v = -\frac{3}{5}\)
02

Use the Formula for Tangent of Difference-of-Angles

Next, substitute the computed values into the formula for the tangent of the difference of two angles which is \(\tan (u-v) = \frac{\sin u \cos v - \cos u \sin v}{\cos u \cos v + \sin u \sin v}\).
03

Calculate the Exact Value

Substitute the computed values \(\sin u = -\frac{7}{25}\), \(\cos u = -\frac{24}{25}\), \(\sin v = -\frac{3}{5}\), and \(\cos v = -\frac{4}{5}\) into the formula. This gives \(\tan (u-v) = \frac{(-\frac{7}{25}*- \frac{4}{5})-(-\frac{24}{25}*-\frac{3}{5})}{(-\frac{24}{25}*- \frac{4}{5}) + (-\frac{7}{25}*-\frac{3}{5})} = \frac{56}{105}\).

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