| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Quadratic in sin²/cos²/tan² |
| Difficulty | Standard +0.3 Part (a) is trivial algebraic manipulation to find tan x. Part (b) applies this result directly. Part (c) requires the Pythagorean identity to convert to a quadratic in cos y, then solving and checking validity - this is a standard C2 technique but involves multiple steps. Overall slightly above average difficulty for C2 due to the quadratic trigonometric equation, but still a routine textbook exercise. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\tan x = \frac{8}{3}\) (or exact equivalent, or 3 s.f. or better) | B1 | (1 mark) |
| (b) \(\tan x = \frac{8}{3}\) \(x = 69.4°\) (α), \(x = 249.4°\) (180 + α) | M1, A1, A1ft | (3 marks) |
| (c) \(3(1 - \cos^2 y) - 8\cos y = 0\) \(3\cos^2 y + 8\cos y - 3 = 0\) | M1, A1 | |
| \((3\cos y - 1)(\cos y + 3) = 0\) \(\cos y = ...\), \(\frac{1}{3}\) (or–3) | M1, A1 | |
| \(y = 70.5°\) (β), \(x = 289.5°\) (360 – β) | A1, A1ft | (6 marks) |
**(a)** $\tan x = \frac{8}{3}$ (or exact equivalent, or 3 s.f. or better) | B1 | (1 mark)
**(b)** $\tan x = \frac{8}{3}$ $x = 69.4°$ (α), $x = 249.4°$ (180 + α) | M1, A1, A1ft | (3 marks)
**(c)** $3(1 - \cos^2 y) - 8\cos y = 0$ $3\cos^2 y + 8\cos y - 3 = 0$ | M1, A1 |
$(3\cos y - 1)(\cos y + 3) = 0$ $\cos y = ...$, $\frac{1}{3}$ (or–3) | M1, A1 |
$y = 70.5°$ (β), $x = 289.5°$ (360 – β) | A1, A1ft | (6 marks)\begin{enumerate}[label=(\alph*)]
\item Given that $3 \sin x = 8 \cos x$, find the value of $\tan x$. [1]
\item Find, to 1 decimal place, all the solutions of $3 \sin x - 8 \cos x = 0$ in the interval $0 \leq x < 360°$. [3]
\item Find, to 1 decimal place, all the solutions of $3 \sin^2 y - 8 \cos y = 0$ in the interval $0 \leq y < 360°$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [10]}}