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The complex number (3 + 2i sin(theta)) / (1 - 2i sin(theta)) is purely imaginary when theta equals:

1. n*pi/6, n in I
2. n*pi + pi/4, n in I
3. n*pi +- pi/3, n in I
4. None of these

Correct answer: n*pi +- pi/3, n in I

Solution

Rationalising gives real part (3 - 4 sin² theta)/(1 + 4 sin² theta); setting it to zero gives sin² theta = 3/4, i.e. theta = n*pi +- pi/3.

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