Question

Evaluate \( \cos \left( \cot^{-1} \left( \frac{7}{24} \right) \right) \):

• A. \( \frac{24}{25} \)
• B. \( \frac{7}{24} \)
• C. \( \frac{7}{27} \)
• D. \( \frac{7}{25} \)
• E. \( \frac{24}{27} \)

Hint:

To evaluate inverse trigonometric functions, draw the corresponding right triangle and use the Pythagorean theorem to find the missing side.

Answer 1

The Correct Option is D

We are given \( \cot^{-1} \left( \frac{7}{24} \right) \), which means \( \cot \theta = \frac{7}{24} \). 
From the definition of cotangent, we know: \[ \cot \theta = \frac{{adjacent}}{{opposite}} = \frac{7}{24} \] This means we can form a right triangle with the adjacent side as 7 and the opposite side as 24. 
The hypotenuse \( h \) is given by: \[ h = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \] Now, \( \cos \theta = \frac{{adjacent}}{{hypotenuse}} = \frac{7}{25} \). Thus, the correct answer is \( \frac{7}{25} \).