| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Product of trig functions |
| Difficulty | Standard +0.3 Part (a) requires recognizing that cos θ is bounded between -1 and 1, making both factors impossible to equal zero—a straightforward conceptual check. Part (b) involves factorizing using sin y = tan y · cos y, leading to a simple quadratic in cos y, which is standard AS-level trigonometry. The question tests understanding of trigonometric bounds and basic algebraic manipulation rather than requiring novel problem-solving. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos\theta = \frac{4}{3}\) and \(\cos\theta = -\frac{5}{2}\); since \(\cos\theta\) takes values between -1 and 1, there does not exist such a \(\theta\) to satisfy these equations | B1, B1 | Attempts to find the values of \(\cos\theta\); Requires a correct statement and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 0, \pi\); \(\cos y = -\frac{2}{3}\) gives no solutions; Solutions: \(y = 0, \pi\) | M1, M1, A1 | Uses a trigonometric substitution in the given equation; Correct work leading to 2 factors; All solutions correct |
### Part a:
$\cos\theta = \frac{4}{3}$ and $\cos\theta = -\frac{5}{2}$; since $\cos\theta$ takes values between -1 and 1, there does not exist such a $\theta$ to satisfy these equations | B1, B1 | Attempts to find the values of $\cos\theta$; Requires a correct statement and conclusion
### Part b:
$y = 0, \pi$; $\cos y = -\frac{2}{3}$ gives no solutions; Solutions: $y = 0, \pi$ | M1, M1, A1 | Uses a trigonometric substitution in the given equation; Correct work leading to 2 factors; All solutions correct\begin{enumerate}[label=(\alph*)]
\item Explain mathematically why there are no values of $\theta$ that satisfy the equation
$$(3\cos\theta - 4)(2\cos\theta + 5) = 0$$ [2]
\item Giving your solutions to one decimal place, where appropriate, solve the equation
$$3\sin y + 2\tan y = 0 \quad \text{for } 0 \leq y \leq \pi$$
(Solutions based entirely on graphical or numerical methods are not acceptable.) [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q12 [5]}}