Question

If \[ \sin\theta=\frac{3}{5}, \] then \(\cos\theta=\)

• A. \(\frac{4}{5}\) but not \(-\frac{4}{5}\)
• B. \(-\frac{4}{5}\) or \(\frac{4}{5}\)
• C. \(-\frac{4}{5}\) but not \(\frac{4}{5}\)
• D. \(\frac{3}{5}\) but not \(-\frac{3}{5}\)

Hint:

If quadrant is not given, both positive and negative values may be possible after taking square root.

Answer 1

The Correct Option is B

Concept:
Use the identity: \[ \sin^2\theta+\cos^2\theta=1 \]

Step 1: Given: \[ \sin\theta=\frac{3}{5} \]

Step 2: Square both sides: \[ \sin^2\theta=\frac{9}{25} \]

Step 3: Use identity: \[ \cos^2\theta=1-\sin^2\theta \] \[ \cos^2\theta=1-\frac{9}{25} \] \[ \cos^2\theta=\frac{16}{25} \]

Step 4: Taking square root: \[ \cos\theta=\pm \frac{4}{5} \]

Step 5: Since quadrant is not specified, \(\cos\theta\) may be positive or negative. \[ \boxed{-\frac{4}{5}\ \text{or}\ \frac{4}{5}} \]