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In the first quadrant, let A1 be the area of the region bounded by y = sin(x), y = cos(x) and the y-axis, and let A2 be the area of the region bounded by y = sin(x), y = cos(x), the x-axis and the line x = pi/2. Which relation holds?

1. A1: A2 = 1: sqrt(2) and A1 + A2 = 1
2. A1 = A2 and A1 + A2 = sqrt(2)
3. 2*A1 = A2 and A1 + A2 = 1 + sqrt(2)
4. A1: A2 = 1: 2 and A1 + A2 = sqrt(2) - 1

Correct answer: A1 = A2 and A1 + A2 = sqrt(2)

Solution

By the symmetry of sin and cos about x = pi/4, the two regions have equal area, and their sum works out to sqrt(2).

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