The sum of all values of θ ∈[0,2 π] satisfying 2 sin ² θ=cos 2 θ and 2 cos ² θ=3 sin θ is

Solution

Maths sir · 4 steps

Turn both equations into equations in sin θ, then keep only the values that satisfy both.

1. First equation. Write cos 2θ = 1 - 2 sin² θ. Then 2 sin² θ = 1 - 2 sin² θ, so 4 sin² θ = 1 and sin θ = ±1/2.
2. Second equation. Write cos² θ = 1 - sin² θ. This gives a quadratic in sin θ.

   2 sin² θ + 3 sin θ - 2 = 0 ⇒ (2 sin θ - 1)(sin θ + 2) = 0

3. Reject one root. sin θ = -2 is impossible, so sin θ = 1/2. This also satisfies the first equation, while sin θ = -1/2 fails the second.
4. So. In [0, 2π], sin θ = 1/2 gives θ = π/6 and 5π/6. Their sum is π, option (a).

The idea to keep. When two trig equations must hold together, solve each one for the same function, take the common values, and only then list the angles in the given interval.

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