Question

If \( \cos A = \frac{4}{5} \), then the value of \( \tan A \) is :

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

Hint:

Memorizing common Pythagorean triplets like (3, 4, 5) helps solve these problems instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a right-angled triangle, \( \cos A = \frac{\text{Base}}{\text{Hypotenuse}} \) and \( \tan A = \frac{\text{Perpendicular}}{\text{Base}} \).
Step 2: Key Formula or Approach:
Use Pythagoras Theorem: \( P^2 + B^2 = H^2 \).
Step 3: Detailed Explanation:
Let \( \text{Base} = 4k \) and \( \text{Hypotenuse} = 5k \).
\[ \text{Perpendicular}^2 = (5k)^2 - (4k)^2 \]
\[ P^2 = 25k^2 - 16k^2 = 9k^2 \implies P = 3k \]
Now, \( \tan A = \frac{P}{B} = \frac{3k}{4k} = \frac{3}{4} \).
Step 4: Final Answer:
The value is \( \frac{3}{4} \).