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

Step 1: Understanding the given expression:
We are given the expression \( \sin^2 \theta + \sin \theta + \cos^2 \theta \) and are asked to find the value of \( \theta \) for which this expression equals 2.

Step 2: Using the Pythagorean identity:
Recall the Pythagorean identity for trigonometric functions:
\[ \sin^2 \theta + \cos^2 \theta = 1 \] Using this identity, we can simplify the given expression:
\[ \sin^2 \theta + \sin \theta + \cos^2 \theta = 1 + \sin \theta \] Now, the expression becomes:
\[ 1 + \sin \theta = 2 \]

Step 3: Solving for \( \sin \theta \):
To solve for \( \sin \theta \), subtract 1 from both sides of the equation:
\[ \sin \theta = 2 - 1 \] \[ \sin \theta = 1 \]

Step 4: Finding the value of \( \theta \):
We know that \( \sin \theta = 1 \) when \( \theta = 90^\circ \) or \( \theta = \frac{\pi}{2} \) radians.

Step 5: Conclusion:
Therefore, the value of \( \theta \) for which \( \sin^2 \theta + \sin \theta + \cos^2 \theta = 2 \) is \( \theta = 90^\circ \) or \( \theta = \frac{\pi}{2} \) radians.