Question

If $\sin \theta \cos \theta>0$, then $\theta$ lies

• A. only in the first quadrant
• B. only in the second quadrant
• C. in the first quadrant or in the fourth quadrant
• D. in the second quadrant or in the fourth quadrant
• E. in the first quadrant or in the third quadrant

Hint:

If product of trig functions is positive, both must have same sign.

Answer 1

Answer: (E)

Concept:
• $\sin\theta \cos\theta > 0$ means both $\sin\theta$ and $\cos\theta$ have same sign

Step 1: Check signs in quadrants

• First quadrant: $\sin\theta > 0$, $\cos\theta > 0$ $\Rightarrow$ product $> 0$
• Second quadrant: $\sin\theta > 0$, $\cos\theta < 0$ $\Rightarrow$ product $< 0$
• Third quadrant: $\sin\theta < 0$, $\cos\theta < 0$ $\Rightarrow$ product $> 0$
• Fourth quadrant: $\sin\theta < 0$, $\cos\theta > 0$ $\Rightarrow$ product $< 0$

Step 2: Identify valid quadrants
Product is positive in: \[ \text{First and Third quadrants} \]
Final Conclusion:
Option (E)