Question

If \(\sin \theta = \frac{3}{5}\) and \(\cos \theta < 0\), then the value of \(\tan \theta\) is

• A. \(-\frac{3}{4}\)
• B. \(-\frac{3}{5}\)
• C. \(\frac{3}{4}\)
• D. \(-\frac{4}{3}\)
• E. \(\frac{4}{3}\)

Hint:

Remember the "ASTC" rule: All (Q1), Sine (Q2), Tan (Q3), Cosine (Q4) are positive. If Sine is positive and Cosine is negative, you are in Q2, where only Sine is positive. So Tan must be negative.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The signs of trigonometric functions depend on the quadrant in which the angle \(\theta\) lies.
- \(\sin \theta > 0\) in Quadrants I and II.
- \(\cos \theta < 0\) in Quadrants II and III.
Therefore, \(\theta\) lies in Quadrant II.

Step 2: Key Formula or Approach:
1. In Quadrant II, \(\tan \theta\) is negative.
2. Use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

Step 3: Detailed Explanation:
Given \(\sin \theta = \frac{3}{5}\).
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left( \frac{3}{5} \right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Since \(\cos \theta < 0\), we have \(\cos \theta = -\frac{4}{5}\).
Now, calculate \(\tan \theta\):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/5}{-4/5} = -\frac{3}{4} \]

Step 4: Final Answer:
The value of \(\tan \theta\) is \(-\frac{3}{4}\).