If the real part of the complex number z = (3 + 2i cos θ)/(1 - 3i cos θ),θ ∈ ( 0,π/2 ) is zero, then the value of sin²3θ + cos²θ is equal to _______________.

Solution

Maths sir · 4 steps

Multiply top and bottom by the conjugate of the denominator, then set the real part to zero.

1. Rationalise. Write c = cos θ for short. Multiply numerator and denominator by 1 + 3ic, the conjugate of the denominator.

   z = (3 + 2ic)(1 + 3ic)/(1 + 9c²) = ((3 - 6c²) + 11ic)/(1 + 9c²)

2. Real part zero. The denominator 1 + 9c² is always positive, so the real part is zero only when 3 - 6c² = 0. That gives cos² θ = 1/2.
3. Find theta. In (0, π/2), cos θ = 1/√2, so θ = π/4 and 3θ = 3π/4. Then sin² 3θ = (1/√2)² = 1/2.
4. So. sin² 3θ + cos² θ = 1/2 + 1/2 = 1.

The idea to keep. To read off the real or imaginary part of a fraction, first make the denominator real by multiplying with its conjugate. Then compare parts of the numerator only.

Answer from the NTA final answer key. Solution by PrepWiser.