Problem
Evaluate: \( \tan\left(\cos^{-1}\left(\frac{-7}{25}\right)\right) \)
Solution
Let \( \theta = \cos^{-1}\left(\frac{-7}{25}\right) \)
Then:
\[ \cos \theta = \frac{-7}{25} = \frac{\text{Base}}{\text{Hypotenuse}} \]
- Base = -7
- Hypotenuse = 25
Perpendicular:
\[ \sqrt{25^2 – 7^2} = \sqrt{625 – 49} = \sqrt{576} = 24 \]
Now,
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{24}{-7} \]
Therefore:
\[ \tan\left(\cos^{-1}\left(\frac{-7}{25}\right)\right) = -\frac{24}{7} \]
Final Answer
\[ \boxed{-\frac{24}{7}} \]
Explanation
Since cos⁻¹x for negative x lies in the second quadrant, tangent is negative.