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Chapter 5: Problem 53

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

Short Answer

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

Step by step solution

01

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, \( \sin v = -\sqrt{1 - \cos^2v} = -\sqrt{1 - \left(-\dfrac{4}{5} \right)^2} = -\dfrac{3}{5} \).
02

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, \( \tan(u - v) = \dfrac{\left(-\dfrac{7}{25}\right) \left(-\dfrac{4}{5}\right) - \left(-\dfrac{24}{25}\right) \left(-\dfrac{3}{5}\right)}{\left(-\dfrac{24}{25}\right) \left(-\dfrac{4}{5}\right) + \left(-\dfrac{7}{25}\right) \left(-\dfrac{3}{5}\right)} \).
03

Simplify the expression

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

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Most popular questions from this chapter

In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \sec u = - 2, \dfrac{\pi}{2} < u < \pi \)

In Exercises 111 - 124, verify the identity. \( \tan \dfrac{u}{2} = \csc u - \cot u \)

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( 112^\circ 30^\prime \)

In Exercises 73-76, use the half-angle formulas to simplify the expression. \( \sqrt{\dfrac{1 - \cos 6x}{2}} \)

In Exercises 77-80, find all solutions of the equation in the interval \( [0, 2\pi) \). Use a graphing utility to graph the equation and verify the solutions. \( \sin \dfrac{x}{2} + \cos x = 0 \)
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