| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Moderate -0.3 Part (a) requires routine algebraic manipulation (combining fractions over a common denominator) and applying the standard identity sin²θ + cos²θ = 1, which is straightforward recall. Part (b) is a direct substitution using the result from (a), followed by solving a basic trigonometric equation with multiple solutions in a given interval. While multi-step, all techniques are standard C3 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
8
\begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$\frac { 1 } { 1 + \cos \theta } + \frac { 1 } { 1 - \cos \theta } = 32$$
can be written in the form
$$\operatorname { cosec } ^ { 2 } \theta = 16$$
\item Hence, or otherwise, solve the equation
$$\frac { 1 } { 1 + \cos ( 2 x - 0.6 ) } + \frac { 1 } { 1 - \cos ( 2 x - 0.6 ) } = 32$$
giving all values of $x$ in radians to two decimal places in the interval $0 < x < \pi$.\\
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q8 [9]}}