| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Quadratic in sin²/cos²/tan² |
| Difficulty | Moderate -0.3 This is a straightforward trigonometric equation requiring standard techniques: dividing by cos²θ to obtain tan²θ = 5/4, then solving for tan θ and finding angles in the given range. The method is routine for AS-level students and involves no novel insight, making it slightly easier than average but still requiring competent algebraic manipulation and understanding of the tan function's periodicity. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks |
|---|---|
| 4(a) | Uses a trig identity, either |
| Answer | Marks | Guidance |
|---|---|---|
| a single trig function. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| sin θ or cos θ | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| OE Must be in exact form | 1.1b | A1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 4(b) | Obtains at least two correct |
| Answer | Marks | Guidance |
|---|---|---|
| OE | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| AWRT 48, 132. 228, 312 | 1.1b | A1 |
| Subtotal | 2 | |
| Question 4 Total | 5 | |
| Q | Marking instructions | AO |
Question 4:
--- 4(a) ---
4(a) | Uses a trig identity, either
s i n
t a n =
c o s
or
sin2+cos2=1
correctly to obtain an equation in
a single trig function. | 1.1a | M1 | 5 cos2θ = 4 sin2θ
5 sin2
= = tan2θ
cos2
4
5
tanθ = ±
2
5 5
Obtains tan2= or s i n 2 =
4 9
4
or c o s 2 =
9
PI by one correct value for tan θ
sin θ or cos θ | 1.1b | A1
5
Obtains tan θ = ±
2
OE Must be in exact form | 1.1b | A1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 4(b) ---
4(b) | Obtains at least two correct
solutions in the range based on
the value of their tan θ , sin θ or
cos θ
OE | 1.1a | M1 | θ = 48.2, 131.8. 228.2, 311.8
Obtains all 4 correct solutions
and no further ones
AWRT 48, 132. 228, 312 | 1.1b | A1
Subtotal | 2
Question 4 Total | 5
Q | Marking instructions | AO | Marks | Typical solutionIt is given that $5\cos^2 \theta - 4\sin^2 \theta = 0$
\begin{enumerate}[label=(\alph*)]
\item Find the possible values of $\tan \theta$, giving your answers in exact form.
[3 marks]
\item Hence, or otherwise, solve the equation
$$5\cos^2 \theta - 4\sin^2 \theta = 0$$
giving all solutions of $\theta$ to the nearest $0.1°$ in the interval $0° \leq \theta \leq 360°$
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2023 Q4 [5]}}