| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Show then solve substituted equation |
| Difficulty | Standard +0.3 Part (i) is a standard algebraic manipulation using tan x = sin x/cos x and the Pythagorean identity, requiring routine substitution and rearrangement. Part (ii) applies the result with a compound angle substitution (x = 2θ) and solving a quadratic in cos x, then finding angles in a specified range. This is slightly above average due to the multi-step nature and range considerations, but follows well-established procedures without requiring novel insight. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\sin x\left(\frac{\sin x}{\cos x}\right) = \cos x + 5\) | M1 | AO3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| \(2(1 - \cos^2 x) = \cos^2 x + 5\cos x\) | M1 | AO3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| \(3\cos^2 x + 5\cos x - 2 = 0\) | A1 | AO2.1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((3\cos 2\theta - 1)(\cos 2\theta + 2) = 0\) | M1 | AO1.1a |
| \(\cos 2\theta = \frac{1}{3}\) (and \(\cos 2\theta = -2\)) | A1 | AO1.1 |
| \(\theta = \frac{1}{2}\arccos\left(\frac{1}{3}\right)\) | M1 | AO1.1 |
| \(\theta = 35.3°\) | A1 | AO1.1 |
| \(\theta = 144.7°\) | A1 | AO1.1 |
## Question 7:
### Part (i):
$2\sin x\left(\frac{\sin x}{\cos x}\right) = \cos x + 5$ | **M1** | AO3.1a | Uses $\tan x = \sin x / \cos x$
$2\sin^2 x = \cos^2 x + 5\cos x$
$2(1 - \cos^2 x) = \cos^2 x + 5\cos x$ | **M1** | AO3.1a | Uses $\sin^2 x = 1 - \cos^2 x$
$2 - 2\cos^2 x = \cos^2 x + 5\cos x$
$3\cos^2 x + 5\cos x - 2 = 0$ | **A1** | AO2.1 | AG – correct working throughout | Must show sufficient working to justify the given answer
### Part (ii):
$(3\cos 2\theta - 1)(\cos 2\theta + 2) = 0$ | **M1** | AO1.1a | Attempt to solve 3-term quadratic
$\cos 2\theta = \frac{1}{3}$ (and $\cos 2\theta = -2$) | **A1** | AO1.1 | Condone $\cos x = \frac{1}{3}$
$\theta = \frac{1}{2}\arccos\left(\frac{1}{3}\right)$ | **M1** | AO1.1 | Correct order of operation to find one value of $\theta$ (or both values of $2\theta$ correct); $(2\theta =)\ 70.52877...,\ 289.471...$
$\theta = 35.3°$ | **A1** | AO1.1 | One correct value to the nearest integer or better
$\theta = 144.7°$ | **A1** | AO1.1 | Cao (35.3 and 144.7) | Any additional values in the range loses final A mark if earned
---7 (i) Show that the equation
$$2 \sin x \tan x = \cos x + 5$$
can be expressed in the form
$$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
(ii) Hence solve the equation
$$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$
giving all values of $\theta$ between $0 ^ { \circ }$ and $180 ^ { \circ }$, correct to 1 decimal place.
\hfill \mbox{\textit{OCR PURE Q7 [8]}}