Question

If tan A = \(\frac{4}{3}\), find sin A and cos A.

Hint:

Memorize Pythagorean triplets like (3, 4, 5), (5, 12, 13) to solve these problems instantly.

Answer 1

Step 1: Understanding the Concept:
Trigonometric ratios are based on the sides of a right triangle. If one ratio is known, others can be found using the Pythagorean theorem.
Step 2: Key Formula or Approach:
1. \( \tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{P}{B} \)
2. \( H = \sqrt{P^2 + B^2} \)
3. \( \sin A = \frac{P}{H}, \cos A = \frac{B}{H} \)
Step 3: Detailed Explanation:
Given \( \tan A = \frac{4}{3} \).
Let Perpendicular (\( P \)) \( = 4k \) and Base (\( B \)) \( = 3k \).
Hypotenuse (\( H \)) \( = \sqrt{(4k)^2 + (3k)^2} = \sqrt{16k^2 + 9k^2} = \sqrt{25k^2} = 5k \).
Now, calculate \( \sin A \):
\[ \sin A = \frac{P}{H} = \frac{4k}{5k} = \frac{4}{5} \]
Calculate \( \cos A \):
\[ \cos A = \frac{B}{H} = \frac{3k}{5k} = \frac{3}{5} \]
Step 4: Final Answer:
\( \sin A = \frac{4}{5} \) and \( \cos A = \frac{3}{5} \).