If m and n respectively are the numbers of positive and negative values of θ in the interval [-π,π] that satisfy the equation cos 2θ cos θ/2 = cos 3θ cos 9θ/2, then mn is equal to ____________.

Solution

Maths sir · 4 steps

Turn both products into sums, and the cos 3θ/2 terms cancel, leaving one clean equation.

1. Product to sum. Use 2 cos A cos B = cos(A+B) + cos(A-B) on each side. The left side gives cos 5θ/2 + cos 3θ/2 and the right side gives cos 15θ/2 + cos 3θ/2.

   cos 15θ/2 = cos 5θ/2 ⇒ 15θ/2 = 2kπ ± 5θ/2, k ∈ ℤ

2. Solve both cases. With the plus sign, 5θ = 2kπ, so θ = 2kπ/5. With the minus sign, 10θ = 2kπ, so θ = kπ/5. The first set is already inside the second, so every solution is θ = kπ/5.
3. Count in the interval. -π ≤ kπ/5 ≤ π gives k = -5, -4, …, 5. The values k = 1 to 5 are positive, so m = 5. The values k = -5 to -1 are negative, so n = 5. The value θ = 0 is neither positive nor negative.
4. So. mn = 5 × 5 = 25.

The common slip: 16. Leaving out the endpoints θ = ±π gives 4 positive and 4 negative values, and so 16. The interval is closed, and both endpoints satisfy the equation.

The idea to keep. When both sides are products of cosines, convert each to a sum first. Matching terms often cancel and leave cos α = cos β, which has the general solution α = 2kπ ± β.

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