Question

The value of \( \cos \left( \tan^{-1} \left( \frac{3}{4} \right) \right) \) is:

• A. \( \frac{4}{5} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{3}{4} \)
• D. \( \frac{2}{5} \)
• E. \( 0 \)

Hint:

Recognizing Pythagorean triples like (3, 4, 5) allows you to bypass the manual calculation of the hypotenuse in many inverse trigonometry problems.

Answer 1

The Correct Option is A

Concept: To evaluate a trigonometric function of an inverse trigonometric function, we can represent the inverse function as an angle \( \theta \) in a right-angled triangle. Since \( \tan^{-1}(\text{ratio}) \) represents the angle whose tangent is that ratio, we can identify the perpendicular and base of the triangle and then find the hypotenuse to calculate the cosine.

Step 1: Represent the inverse tangent as an angle.
Let \( \theta = \tan^{-1} \left( \frac{3}{4} \right) \). This implies that \( \tan \theta = \frac{3}{4} \). In a right-angled triangle, \( \tan \theta = \frac{\text{Perpendicular (P)}}{\text{Base (B)}} \). So, \( P = 3 \) and \( B = 4 \).

Step 2: Calculate the hypotenuse.
Using the Pythagorean theorem \( H^2 = P^2 + B^2 \): \[ H = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 3: Evaluate the cosine of the angle.
We need to find \( \cos \theta \). In a right-angled triangle, \( \cos \theta = \frac{\text{Base (B)}}{\text{Hypotenuse (H)}} \). \[ \cos \theta = \frac{4}{5} \]