Standard +0.3 This is a standard trigonometric equation requiring the Pythagorean identity to convert to a quadratic in sin θ, then solving the quadratic and finding angles in the given range. It's slightly above routine due to the need to recognize one solution is invalid (sin θ > 1) and find two angles for the valid solution, but remains a textbook exercise with no novel insight required.
Find all the solutions of the equation
$$\cos^2 \theta = 10 \sin \theta + 4$$
for \(0° < \theta < 360°\), giving your answers to the nearest degree.
Fully justify your answer.
[5 marks]
Question 4:
4 | Uses identity cos2θ = 1– sin2θ | 1.2 | B1 | 1 – sin2θ = 10sinθ +4
0 = sin2θ + 10sinθ +3
sinθ = –0.3095 or –9.6904
sinθ = –9.6904 (not valid)
sin–1(–0.3095) = –18.03°
θ = 198° or θ = 342°
Solves their quadratic in sinθ
PI by at least one correct value
of θ or - 18º | 1.1a | M1
Explains that the second
solution or both solutions is/are
inappropriate.
Accept N/A, out of range, no
solutions, math error, reject OE
Do not accept a ‘×’ or –9.69
(OE) crossed out alone. | 2.4 | E1F
Obtains one correct value for θ
AWRT 198° or 342° | 1.1b | A1
Obtains two correct values for θ
Condone - 18º included, but no
other answers. | 1.1b | A1
Question 4 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
Find all the solutions of the equation
$$\cos^2 \theta = 10 \sin \theta + 4$$
for $0° < \theta < 360°$, giving your answers to the nearest degree.
Fully justify your answer.
[5 marks]
\hfill \mbox{\textit{AQA AS Paper 1 2022 Q4 [5]}}