| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Disprove statement by counterexample |
| Difficulty | Moderate -0.3 Part (a) requires finding a simple counterexample (e.g., θ = π/2 where cosec θ - sin θ = 0), which is straightforward. Part (b) involves algebraic manipulation to form a quadratic in sin θ and solving, which is standard C3 technique but requires careful execution. Overall slightly easier than average due to the guided structure and routine methods. |
| Spec | 1.01c Disproof by counter example1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) If \(\theta = \frac{\pi}{2}\), \(\sin \theta = 1\), \(\cosec \theta = 1\) | M1 | |
| \(\therefore \cosec \theta - \sin \theta = 1 - 1 = 0\) | M1 | |
| \(\therefore\) statement is false | A1 | |
| (b) \(1 - \sin^2 \theta = 2 \sin \theta\) | M1 | |
| \(\sin^2 \theta + 2 \sin \theta - 1 = 0\) | M1 | |
| \(\sin \theta = \frac{-2 \pm \sqrt{4+4}}{2} = -1 - \sqrt{2}\) (no solutions) or \(-1 + \sqrt{2}\) | M1 A1 | |
| \(\theta = 0.4271, \pi - 0.4271\) | M1 | |
| \(\theta = 0.43, 2.71\) (2dp) | A1 | (7 marks) |
**(a)** If $\theta = \frac{\pi}{2}$, $\sin \theta = 1$, $\cosec \theta = 1$ | M1 |
$\therefore \cosec \theta - \sin \theta = 1 - 1 = 0$ | M1 |
$\therefore$ statement is false | A1 |
**(b)** $1 - \sin^2 \theta = 2 \sin \theta$ | M1 |
$\sin^2 \theta + 2 \sin \theta - 1 = 0$ | M1 |
$\sin \theta = \frac{-2 \pm \sqrt{4+4}}{2} = -1 - \sqrt{2}$ (no solutions) or $-1 + \sqrt{2}$ | M1 A1 |
$\theta = 0.4271, \pi - 0.4271$ | M1 |
$\theta = 0.43, 2.71$ (2dp) | A1 | (7 marks)
---\begin{enumerate}
\item (a) Prove, by counter-example, that the statement
\end{enumerate}
$$\text { "cosec } \theta - \sin \theta > 0 \text { for all values of } \theta \text { in the interval } 0 < \theta < \pi \text { " }$$
is false.\\
(b) Find the values of $\theta$ in the interval $0 < \theta < \pi$ such that
$$\operatorname { cosec } \theta - \sin \theta = 2$$
giving your answers to 2 decimal places.\\
\hfill \mbox{\textit{Edexcel C3 Q2 [7]}}