Problem
Evaluate: \( \cos\left(\tan^{-1}\left(\frac{-3}{4}\right)\right) \)
Solution
Let \( \theta = \tan^{-1}\left(\frac{-3}{4}\right) \)
Then:
\[ \tan \theta = \frac{-3}{4} = \frac{\text{Perpendicular}}{\text{Base}} \]
- Perpendicular = -3
- Base = 4
Hypotenuse:
\[ \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 \]
Now,
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{4}{5} \]
Therefore:
\[ \cos\left(\tan^{-1}\left(\frac{-3}{4}\right)\right) = \frac{4}{5} \]
Final Answer
\[ \boxed{\frac{4}{5}} \]
Explanation
tan⁻¹x lies in (−π/2, π/2). For negative x, the angle is in the fourth quadrant where cosine is positive.