Let S be the sum of all solutions (in radians) of the equation sin ⁴θ + cos ⁴θ - sin θ cos θ = 0 in [0, 4π]. Then 8S/π is equal to ____________.

Solution

Maths sir · 4 steps

Write everything in terms of sin 2θ and you get a quadratic that is easy to factorise.

1. Rewrite the powers. sin⁴ θ + cos⁴ θ = (sin² θ + cos² θ)² - 2 sin² θ cos² θ = 1 - 1/2 sin² 2θ. Also sin θ cos θ = 1/2 sin 2θ.

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

2. Pick the valid root. sin 2θ = -2 is impossible, so sin 2θ = 1. That gives 2θ = π/2 + 2nπ, that is, θ = π/4 + nπ.
3. List solutions. In [0, 4π] the solutions are π/4, 5π/4, 9π/4, 13π/4. Their sum is S = 28π/4 = 7π.
4. So. 8S/π = (8 × 7π)/π = 56.

The common slip: 12. Letting 2θ run only over [0, 4π] keeps just π/4 and 5π/4, so S = 3π/2 and the answer comes out as 12, not 56.

The idea to keep. When sin⁴ + cos⁴ appears with sin θ cos θ, switch to sin 2θ. Then remember that θ in [0, 4π] means 2θ runs over [0, 8π].

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