Question

If \( \cos A = \dfrac{3}{5} \), calculate \( \sin A \) and \( \tan A \).

Hint:

Remember: If one trigonometric ratio is given, use identities like \( \sin^2 A + \cos^2 A = 1 \) to find the others.

Answer 1

Step 1: Use the identity between sine and cosine.}
We know that \[ \sin^2 A + \cos^2 A = 1 \] Given that \[ \cos A = \frac{3}{5} \] So, \[ \cos^2 A = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
Step 2: Find the value of \( \sin^2 A \).}
Substituting in the identity, we get \[ \sin^2 A = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Step 3: Find the value of \( \sin A \).}
Taking the positive square root for an acute angle, \[ \sin A = \sqrt{\frac{16}{25}} = \frac{4}{5} \]
Step 4: Find the value of \( \tan A \).}
Now, \[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]
Step 5: Write the final answer.}
Therefore, \[ \sin A = \frac{4}{5}, \qquad \tan A = \frac{4}{3} \]