Question

For what value of \(\theta\), \(\sin^2\theta + \sin\theta + \cos^2\theta\) is equal to 2?

• A. \(45^\circ\)
• B. \(0^\circ\)
• C. \(90^\circ\)
• D. \(30^\circ\)

Answer 1

The Correct Option is C

Problem:
We are given the trigonometric expression:
\[ \sin^2\theta + \sin\theta + \cos^2\theta \]
We are to find the value of \( \theta \) for which the expression equals 2.

Step 1: Use trigonometric identity
Recall the fundamental identity:
\[ \sin^2\theta + \cos^2\theta = 1 \]
So we substitute this into the given expression:
\[ \sin^2\theta + \cos^2\theta + \sin\theta = 1 + \sin\theta \]

Step 2: Set the expression equal to 2
\[ 1 + \sin\theta = 2 \Rightarrow \sin\theta = 2 - 1 = 1 \]

Step 3: Solve for \( \theta \)
\[ \sin\theta = 1 \Rightarrow \theta = 90^\circ \quad \text{(or in radians: } \frac{\pi}{2} \text{)} \]
This is because sine of 90° is exactly 1.

Final Answer:
The required value of \( \theta \) is \(90^\circ\) or \(\frac{\pi}{2}\) radians.