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

To find the value of \( \tan A \) given that \( \cos A = \frac{4}{5} \), we will use the Pythagorean identity and the definition of tangent.

1. Recall the Pythagorean identity for trigonometric functions: \(\sin^2 A + \cos^2 A = 1\).
2. Substitute the value of \( \cos A \) into the identity: \(\sin^2 A + \left(\frac{4}{5}\right)^2 = 1\).
3. Calculate: \(\sin^2 A + \frac{16}{25} = 1\).
4. Solve for \( \sin^2 A \): \(\sin^2 A = 1 - \frac{16}{25} = \frac{9}{25}\).
5. Find \( \sin A \) (since \( \sin A \) should be positive in the first quadrant): \(\sin A = \frac{3}{5}\).
6. Given that \( \tan A = \frac{\sin A}{\cos A} \), substitute the values of \( \sin A \) and \( \cos A \): \(\tan A = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}\).
7. Therefore, the correct answer is \(\frac{3}{4}\).

The value of \( \tan A \) when \( \cos A = \frac{4}{5} \) is \(\frac{3}{4}\).