Question:
If \( x \in [0, \frac{\pi}{2}] \), \( y \in [0, \frac{\pi}{2}] \) and \( \sin x + \cos y = 2 \), then the value of \( x + y \) is equal to:
If \( x \in [0, \frac{\pi}{2}] \), \( y \in [0, \frac{\pi}{2}] \) and \( \sin x + \cos y = 2 \), then the value of \( x + y \) is equal to:
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Whenever you see an equation where a sum of trigonometric functions equals the sum of their maximum possible values (like \( \sin A + \sin B = 2 \)), treat each term as equal to 1 to find the angles quickly.
Updated On: May 1, 2026
- \( 2\pi \)
- \( \pi \)
- \( \frac{\pi}{4} \)
- \( \frac{\pi}{2} \)
- \( 0 \)
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The Correct Option is D
Solution and Explanation
Concept: The range of the sine and cosine functions is restricted to the interval \( [-1, 1] \). For the sum of two such functions to reach their combined maximum possible value (in this case, 2), each individual function must simultaneously equal its own maximum value of 1.
Step 1: Determine the values of \( x \) and \( y \).
Given \( \sin x + \cos y = 2 \), and knowing that \( \sin x \leq 1 \) and \( \cos y \leq 1 \), the only possibility is: \[ \sin x = 1 \quad \text{and} \quad \cos y = 1 \] Within the restricted interval \( [0, \frac{\pi}{2}] \):
• \( \sin x = 1 \implies x = \frac{\pi}{2} \)
• \( \cos y = 1 \implies y = 0 \)
Step 2: Calculate the sum \( x + y \).
\[ x + y = \frac{\pi}{2} + 0 = \frac{\pi}{2} \]
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