Moderate -0.3 This is a straightforward trigonometric manipulation requiring the double angle formula sin(2θ) = 2sin(θ)cos(θ). Part (a) involves algebraic rearrangement (multiply by sin θ, recognize the double angle) with clear guidance on the target form. Part (b) is routine inverse trigonometry. The question is slightly easier than average because it's highly scaffolded and uses standard techniques without requiring problem-solving insight.
2. (a) Show that the equation
$$8 \cos \theta = 3 \operatorname { cosec } \theta$$
can be written in the form
$$\sin 2 \theta = k$$
where \(k\) is a constant to be found.
(b) Hence find the smallest positive solution of the equation
$$8 \cos \theta = 3 \operatorname { cosec } \theta$$
giving your answer, in degrees, to one decimal place.
States or uses \(\cosec\theta = \frac{1}{\sin\theta}\)
B1
May be implied by line \(8\sin\theta\cos\theta = 3\)
Attempts to use \(\sin 2\theta = 2\sin\theta\cos\theta\)
M1
Proceeds to \(\sin 2\theta = k\) where \(
\(\sin 2\theta = \frac{3}{4}\) cso
A1
Achieves \(\sin 2\theta = \frac{3}{4}\) o.e. with no errors
Question 2(b):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
Correct order of operations to find \(\theta\), look for \(\dfrac{\arcsin\!\left(\frac{3}{4}\right)}{2}\)
M1
Uses correct order of operations to find any value of \(\theta\) in degrees or radians that works for their \(\sin 2\theta = k\), with \(
\(\theta =\) awrt \(24.3°\)
A1
\(\theta =\) awrt \(24.3°\) ONLY
## Question 2(a):
$8\cos\theta = 3\cosec\theta$
| Answer/Working | Marks | Guidance |
|---|---|---|
| States or uses $\cosec\theta = \frac{1}{\sin\theta}$ | B1 | May be implied by line $8\sin\theta\cos\theta = 3$ |
| Attempts to use $\sin 2\theta = 2\sin\theta\cos\theta$ | M1 | Proceeds to $\sin 2\theta = k$ where $|k| \leq 1$. Note: $8\sin\theta\cos\theta = 3 \Rightarrow 8\sin 2\theta = 3$ would be M0 |
| $\sin 2\theta = \frac{3}{4}$ cso | A1 | Achieves $\sin 2\theta = \frac{3}{4}$ o.e. with no errors |
## Question 2(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct order of operations to find $\theta$, look for $\dfrac{\arcsin\!\left(\frac{3}{4}\right)}{2}$ | M1 | Uses correct order of operations to find any value of $\theta$ in degrees or radians that works for their $\sin 2\theta = k$, with $|k|\leq 1$ |
| $\theta =$ awrt $24.3°$ | A1 | $\theta =$ awrt $24.3°$ ONLY |
2. (a) Show that the equation
$$8 \cos \theta = 3 \operatorname { cosec } \theta$$
can be written in the form
$$\sin 2 \theta = k$$
where $k$ is a constant to be found.\\
(b) Hence find the smallest positive solution of the equation
$$8 \cos \theta = 3 \operatorname { cosec } \theta$$
giving your answer, in degrees, to one decimal place.
\hfill \mbox{\textit{Edexcel P3 2022 Q2 [5]}}