| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Solve equation using Pythagorean identities |
| Difficulty | Standard +0.3 This is a standard C3 trigonometric equation requiring the Pythagorean identity sec²θ = 1 + tan²θ to convert to a quadratic in secθ, then solving and finding angles in the given range. Part (b) is a straightforward substitution. Slightly above average due to the algebraic manipulation and multiple solutions, but follows a well-practiced technique with no novel insight required. |
| 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 |
4
\begin{enumerate}[label=(\alph*)]
\item By using a suitable trigonometrical identity, solve the equation
$$\tan ^ { 2 } \theta = 3 ( 3 - \sec \theta )$$
giving all solutions to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$.
\item Hence solve the equation
$$\tan ^ { 2 } \left( 4 x - 10 ^ { \circ } \right) = 3 \left[ 3 - \sec \left( 4 x - 10 ^ { \circ } \right) \right]$$
giving all solutions to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } < x < 90 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q4 [9]}}