Problem 53 - Precalculus with Limits

Question

In Exercises \(51-56,\) find the exact value of the trigonometric function given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III. \(\tan (u-v)\)

Step-by-Step Solution

Answer

The exact value of \( \tan(u - v) \) is \( -\dfrac{44}{117} \).

Step 1: Determine the values of \(\cos u\) and \(\sin v\)

Since \( u \) and \( v \) are in the third quadrant where both sine and cosine are negative, the values of \( \cos u \) and \( \sin v \) can be determined from the given \( \sin u = -\dfrac{7}{25} \) and \( \cos v = -\dfrac{4}{5} \) using the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \). Solving for \( \cos u \) gives \( \cos u = -\sqrt{1 - \sin^2u} = -\sqrt{1 - \left(-\dfrac{7}{25} \right)^2} = -\dfrac{24}{25} \). Similarly, \( \sin v = -\sqrt{1 - \cos^2v} = -\sqrt{1 - \left(-\dfrac{4}{5} \right)^2} = -\dfrac{3}{5} \).

Step 2: Apply the formula for tangent of difference of angles

The formula for tangent of difference of angles is \( \tan(u - v) = \dfrac{\sin u \cos v - \cos u \sin v}{\cos u \cos v + \sin u \sin v} \). Substituting the given and calculated values from Step 1, \( \tan(u - v) = \dfrac{\left(-\dfrac{7}{25}\right) \left(-\dfrac{4}{5}\right) - \left(-\dfrac{24}{25}\right) \left(-\dfrac{3}{5}\right)}{\left(-\dfrac{24}{25}\right) \left(-\dfrac{4}{5}\right) + \left(-\dfrac{7}{25}\right) \left(-\dfrac{3}{5}\right)} \).

Step 3: Simplify the expression

Simplify the numerator and denominator to get \( \tan(u - v) = \dfrac{\dfrac{28}{125} - \dfrac{72}{125}}{\dfrac{96}{125} + \dfrac{21}{125}} = \dfrac{-44}{117} \).