If the sum of solutions of the system of equations 2 sin ² θ-cos 2 θ=0 and 2 cos ² θ+3 sin θ=0 in the interval [0,2 π] is k π, then k is equal to __________.

Solution

Maths sir · 5 steps

Write both equations in sin θ alone, then keep only the value they agree on.

1. First equation. Use cos 2θ = 1 - 2 sin² θ. Then 2 sin² θ - 1 + 2 sin² θ = 0, so sin² θ = 1/4 and sin θ = ±1/2.
2. Second equation. Use cos² θ = 1 - sin² θ. Then 2 - 2 sin² θ + 3 sin θ = 0, which rearranges to a quadratic in sin θ.

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

3. Common value. sin θ = 2 is impossible, so the second equation needs sin θ = -1/2. This is one of the values from the first equation, so the system holds exactly when sin θ = -1/2.
4. Solve in the interval. In [0, 2π], sin θ = -1/2 in the third and fourth quadrants: θ = 7π/6 and θ = 11π/6.
5. So. The sum is 7π/6 + 11π/6 = 18π/6 = 3π, so k = 3.

The common slip: 4. Adding all four angles from the first equation alone (π/6, 5π/6, 7π/6, 11π/6) gives 4π, so 4. The second equation rules out sin θ = 1/2.

The idea to keep. For a system of trig equations, reduce every equation to the same function, solve each one, and keep only the common values. Only then find the angles in the given interval.

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