Problem
Evaluate: \( \cot\left(\cos^{-1}\left(\frac{3}{5}\right)\right) \)
Solution
Let \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \)
Then:
\[ \cos \theta = \frac{3}{5} = \frac{\text{Base}}{\text{Hypotenuse}} \]
- Base = 3
- Hypotenuse = 5
Perpendicular:
\[ \sqrt{5^2 – 3^2} = \sqrt{25 – 9} = \sqrt{16} = 4 \]
Now, using:
\[ \cot \theta = \frac{\text{Base}}{\text{Perpendicular}} \]
\[ \cot \theta = \frac{3}{4} \]
Therefore:
\[ \cot\left(\cos^{-1}\left(\frac{3}{5}\right)\right) = \frac{3}{4} \]
Final Answer
\[ \boxed{\frac{3}{4}} \]
Explanation
Using right triangle definitions: cos = base/hypotenuse and cot = base/perpendicular.