In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]

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Recognize that the expression is \( \tan(\cos^{-1}(-\frac{4}{5})) \). Here, \( \cos^{-1}(-\frac{4}{5}) \) represents an angle \( \theta \) whose cosine is \( -\frac{4}{5} \). So, set \( \theta = \cos^{-1}(-\frac{4}{5}) \), which means \( \cos \theta = -\frac{4}{5} \).

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this to find \( \sin \theta \) by substituting \( \cos \theta = -\frac{4}{5} \): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \] Then, \( \sin \theta = \pm \frac{3}{5} \).

Determine the correct sign of \( \sin \theta \) by considering the range of \( \theta = \cos^{-1}(-\frac{4}{5}) \). Since \( \cos \theta \) is negative, \( \theta \) lies in the second quadrant where sine is positive. Therefore, \( \sin \theta = \frac{3}{5} \).

Use the definition of tangent in terms of sine and cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{-\frac{4}{5}} \]

Simplify the fraction by dividing the numerators and denominators: \[ \tan \theta = \frac{3}{5} \times \frac{5}{-4} = -\frac{3}{4} \] This gives the exact value of the original expression.

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