| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Moderate -0.8 Two straightforward C2 trig equations requiring standard techniques: (a) isolate sin, find reference angle, apply phase shift; (b) take square root, find tan values. Both are routine textbook exercises with no problem-solving insight needed, making them easier than average A-level questions. |
| Spec | 1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\sin(\theta+30) = \frac{3}{5}\) | B1 (\(\frac{3}{5}\) on RHS) | |
| \(\theta + 30 = 36.9\) | B1 (\(\alpha =\) AWRT 37) | |
| or \(= 143.1\) | M1 (\((180-\alpha)\)) | |
| \(\theta = 6.9, 113.1\) | A1 cao | (4 marks) |
| (b) \(\tan\theta = \pm 2\) or \(\sin\theta = \pm\frac{2}{\sqrt{5}}\) or \(\cos\theta = \pm\frac{1}{\sqrt{5}}\) | B1 | |
| \((\tan\theta = 2 \Rightarrow)\) \(\theta = 63.4\) | B1 (\(\beta =\) AWRT 63.4) | |
| or \(243.4\) | M1 (\((180+\beta)\)) | |
| \((\tan\theta = -2 \Rightarrow)\) \(\theta = 116.6\) | M1 (\((180-\beta)\)) | |
| or \(296.6\) | M1 (\((180+\) their \(116.6)\)) | (5 marks) |
(a) $\sin(\theta+30) = \frac{3}{5}$ | B1 ($\frac{3}{5}$ on RHS)
$\theta + 30 = 36.9$ | B1 ($\alpha =$ AWRT 37)
or $= 143.1$ | M1 ($(180-\alpha)$)
$\theta = 6.9, 113.1$ | A1 cao | (4 marks)
(b) $\tan\theta = \pm 2$ or $\sin\theta = \pm\frac{2}{\sqrt{5}}$ or $\cos\theta = \pm\frac{1}{\sqrt{5}}$ | B1
$(\tan\theta = 2 \Rightarrow)$ $\theta = 63.4$ | B1 ($\beta =$ AWRT 63.4)
or $243.4$ | M1 ($(180+\beta)$)
$(\tan\theta = -2 \Rightarrow)$ $\theta = 116.6$ | M1 ($(180-\beta)$)
or $296.6$ | M1 ($(180+$ their $116.6)$) | (5 marks)
**Guidance:**
- (a) M1 for $180 -$ their first solution. Must be at the correct stage i.e. for $\theta+30$
- (b) ALL M marks in (b) must be for $\theta = \ldots$
- 1st M1 for $180 +$ their first solution
- 2nd M1 for $180 -$ their first solution
- 3rd M1 for $180+$ their 116.6 or $360 -$ their first solution
**Answers Only** can score full marks in both parts
**Not 1 d.p.:** loses A1 in part (a). In (b) all answers are AWRT.
**Ignore extra solutions outside range**
**Radians:** Allow M marks for consistent work with radians only, but all A and B marks for angles must be in degrees. Mixing degrees and radians is M0.
**Total: 9 marks**
---\begin{enumerate}
\item (a) Find all the values of $\theta$, to 1 decimal place, in the interval $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$ for which
\end{enumerate}
$$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$
(b) Find all the values of $\theta$, to 1 decimal place, in the interval $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$ for which
$$\tan ^ { 2 } \theta = 4$$
\hfill \mbox{\textit{Edexcel C2 2006 Q8 [9]}}