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If the points (sinθ, cosθ) and (3,2) are on the same side of the line x+y=1, then the angle θ must lie in which interval?

1. (0, π/2)
2. (0, π)
3. (π/4, π/2)
4. (0, π/4)

Correct answer: (0, π/2)

Solution

For (3,2): 3+2-1 = 4 > 0, so need sin t + cos t > 1, i.e. sqrt(2) sin(t + pi/4) > 1. This holds for t + pi/4 in (pi/4, 3pi/4), giving t in (0, pi/2).

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