Theory question 111
Permutations and Combinations(a) If kP2 = 72, find the value of k.
(b) Solve the equation 2 cos 2θ − 5 cos θ = 3, for 0° ≤ θ ≤ 360°.
Show the model answer
Model answer
(a) k(k − 1) = 72, (k − 9)(k + 8) = 0, k = 9.
(b) 2(2cos²θ − 1) − 5cos θ = 3, so 4cos²θ − 5cos θ − 5 = 0.
cos θ = (5 ± √105)/8 = 1.906 (impossible) or −0.6559.
θ ≈ 131.0° or 229.0°.
Explanation
Use cos 2θ = 2cos²θ − 1 to get a quadratic in cos θ.