Moderate -0.8 This is a straightforward C2 question requiring basic trig identities. Part (a) is immediate division to get tan θ = 5, and part (b) requires using arctan and adding 180° for the second solution. This is simpler than average A-level questions as it involves minimal steps and only routine application of tan θ = sin θ/cos θ with no problem-solving insight needed.
6. (a) Given that \(\sin \theta = 5 \cos \theta\), find the value of \(\tan \theta\).
(b) Hence, or otherwise, find the values of \(\theta\) in the interval \(0 \leqslant \theta < 360 ^ { \circ }\) for which
$$\sin \theta = 5 \cos \theta ,$$
giving your answers to 1 decimal place.
(a) Must be seen explicitly, e.g. \(\tan \theta = \tan^{-1} 5 = 78.7\) or equiv. is B0, unless \(\tan \theta = 5\) is also seen. (b) The M mark may be implied by working in (a). A1ft for \(180 + \alpha\) where \(\alpha \neq k\). Answers in radians would lose both the A marks. Extra answers between 0 and 360: Deduct the final mark. Alternative: Using \(\cos^2 \theta = 1 - \sin^2 \theta\) (or equiv.) and proceeding to \(\sin \theta = k\) (or equiv.): M1 then A marks as in main scheme.
**(a)** $\tan \theta = 5$ | B1 | (1 mark) |
| --- | --- | --- |
| **(b)** $\tan \theta = k$ | M1 | $\theta = \tan^{-1} k$ |
| $\theta = 78.7, 258.7$ | A1, A1ft | (Accept awrt) (3 marks) |
| | (4 marks) | **(a)** Must be seen explicitly, e.g. $\tan \theta = \tan^{-1} 5 = 78.7$ or equiv. is B0, unless $\tan \theta = 5$ is also seen. **(b)** The M mark may be implied by working in (a). A1ft for $180 + \alpha$ where $\alpha \neq k$. Answers in radians would lose both the A marks. Extra answers between 0 and 360: Deduct the final mark. **Alternative:** Using $\cos^2 \theta = 1 - \sin^2 \theta$ (or equiv.) and proceeding to $\sin \theta = k$ (or equiv.): M1 then A marks as in main scheme. |
6. (a) Given that $\sin \theta = 5 \cos \theta$, find the value of $\tan \theta$.\\
(b) Hence, or otherwise, find the values of $\theta$ in the interval $0 \leqslant \theta < 360 ^ { \circ }$ for which
$$\sin \theta = 5 \cos \theta ,$$
giving your answers to 1 decimal place.\\
\hfill \mbox{\textit{Edexcel C2 2006 Q6 [4]}}