| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Standard +0.3 Part (i) requires rearranging to tan x = -7/3 and finding solutions in the specified range—a standard technique. Part (ii) involves using cos²θ + sin²θ = 1 to form a quadratic in cos θ, then solving—slightly more steps but still routine C2 material. Both are textbook-style exercises with no novel insight required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use \(\frac{\sin x}{\cos x} = \tan x\) to give \(\tan x = -\frac{7}{3}\) | M1 | Or complete method to find \(\sin x\) or \(\cos x\) |
| \(\tan x = -\frac{7}{3}\) or \(\sin x = \pm\frac{7}{\sqrt{58}}\) or \(\cos x = \pm\frac{3}{\sqrt{58}}\) | A1 | Ignore \(\cos x = 0\) as extra answer |
| One correct angle in degrees in range | M1 | Need either \(113.2\) or \(293.2\) |
| \(x = 113.2, 293.2\) | A1 | Accept awrt; extra answers in range lose this mark [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(10\cos^2\theta + \cos\theta = 11(1-\cos^2\theta)-9\) | M1 | Replaces \(\sin^2\theta\) by \((1-\cos^2\theta)\) |
| Solves \(21\cos^2\theta + \cos\theta - 2 = 0\) to give \(\cos\theta = \ldots\) | M1 | Collect terms and solve three term quadratic |
| \(\cos\theta = -\frac{1}{3}\) or \(\frac{2}{7}\) | A1 | Both correct values needed |
| \(\theta = 1.91, 4.37, 1.28\) or \(5.00\) (allow \(5\) instead of \(5.00\)) | M1 A1 A1 | M1: uses inverse cosine for at least two correct answers; A1: any two completely correct; A1: all four correct [6] |
# Question 14:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use $\frac{\sin x}{\cos x} = \tan x$ to give $\tan x = -\frac{7}{3}$ | M1 | Or complete method to find $\sin x$ or $\cos x$ |
| $\tan x = -\frac{7}{3}$ or $\sin x = \pm\frac{7}{\sqrt{58}}$ or $\cos x = \pm\frac{3}{\sqrt{58}}$ | A1 | Ignore $\cos x = 0$ as extra answer |
| One correct angle in degrees in range | M1 | Need either $113.2$ or $293.2$ |
| $x = 113.2, 293.2$ | A1 | Accept awrt; extra answers in range lose this mark **[4]** |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $10\cos^2\theta + \cos\theta = 11(1-\cos^2\theta)-9$ | M1 | Replaces $\sin^2\theta$ by $(1-\cos^2\theta)$ |
| Solves $21\cos^2\theta + \cos\theta - 2 = 0$ to give $\cos\theta = \ldots$ | M1 | Collect terms and solve three term quadratic |
| $\cos\theta = -\frac{1}{3}$ or $\frac{2}{7}$ | A1 | Both correct values needed |
| $\theta = 1.91, 4.37, 1.28$ or $5.00$ (allow $5$ instead of $5.00$) | M1 A1 A1 | M1: uses inverse cosine for at least two correct answers; A1: any two completely correct; A1: all four correct **[6]** |
---14. In this question, solutions based entirely on graphical or numerical methods are not acceptable.\\
(i) Solve, for $0 \leqslant x < 360 ^ { \circ }$,
$$3 \sin x + 7 \cos x = 0$$
Give each solution, in degrees, to one decimal place.\\
(ii) Solve, for $0 \leqslant \theta < 2 \pi$,
$$10 \cos ^ { 2 } \theta + \cos \theta = 11 \sin ^ { 2 } \theta - 9$$
Give each solution, in radians, to 3 significant figures.\\
\hfill \mbox{\textit{Edexcel C12 2015 Q14 [10]}}