Solve the equation 2 tan² θ - 3 sec θ = 0 for 0^(°) leqslant θ < 360^(°). Give your answers to one decimal place where appropriate.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for using the identity tan² θ = sec² θ - 1 to get an equation in terms of sec θ only.
2. A1 for 2(sec² θ - 1) - 3 sec θ = 0 ⇒ 2 sec² θ - 3 sec θ - 2 = 0.
3. M1 for attempting to solve the quadratic equation for sec θ (e.g., factorising, quadratic formula).
4. A1 for (sec θ - 2)(2 sec θ + 1) = 0 ⇒ sec θ = 2 or sec θ = -1/2.
5. M1 for converting to cos θ values: cos θ = 1/2 or cos θ = -2. (Ignore cos θ = -2 as it has no solutions).
6. A1 for θ = 60.0^(°) (from cos θ = 1/2). (One d.p. required)
7. A1 for θ = 300.0^(°) (from cos θ = 1/2). (No other solutions for cos θ = 1/2 within range)