Question

If \(\tan\theta = -4/3\), then the value of \(\sin\theta\) is

• A. \(-4/5\) and not \(4/5\)
• B. \(-4/5\) or \(4/5\)
• C. \(4/5\) and not \(-4/5\)
• D. \(1/5\)

Hint:

Quadrant determines sign of trigonometric functions.

Answer 1

The Correct Option is B

To find the value of \( \sin\theta \) given that \( \tan\theta = -\frac{4}{3} \), we start by understanding the relationship between the trigonometric functions. We know the identity: 

\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

Given:

\(\tan\theta = -\frac{4}{3}\)

This means:

\(\frac{\sin\theta}{\cos\theta} = -\frac{4}{3}\)

Let's assume:

• \(\sin\theta = -4k\)
• \(\cos\theta = 3k\)

Using the Pythagorean Identity:

\(\sin^2\theta + \cos^2\theta = 1\)

Substitute the assumed values:

\(((-4k)^2 + (3k)^2 = 1)\)

Solving this gives:

\((16k^2 + 9k^2 = 1)\)

\((25k^2 = 1)\)

\((k^2 = \frac{1}{25})\)

\((k = \pm\frac{1}{5})\)

Therefore,

• \(\sin\theta = -4k = -4 \left(\pm \frac{1}{5}\right)\)
• \(\sin\theta = -\frac{4}{5} \text{ or } \frac{4}{5}\)

Thus, the possible values of \( \sin\theta \) are \(-\frac{4}{5}\) or \(\frac{4}{5}\), which matches the correct option. The correct answer is:

\(-\frac{4}{5}\) or \(\frac{4}{5}\).