4 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\).
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Question 4:
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Answer Marks
Guidance
\(\tan(\theta+45) = \frac{\tan\theta + \tan 45}{1 - \tan\theta\tan 45} = \frac{\tan\theta + 1}{1 - \tan\theta}\) M1 A1
oe using sin/cos
\(\Rightarrow \frac{\tan\theta + 1}{1-\tan\theta} = 1 - 2\tan\theta\) \(\Rightarrow 1 + \tan\theta = (1-2\tan\theta)(1-\tan\theta) = 1 - 3\tan\theta + 2\tan^2\theta\) M1 A1
Multiplying up and expanding. Any correct one line equation.
\(\Rightarrow 0 = 2\tan^2\theta - 4\tan\theta = 2\tan\theta(\tan\theta - 2)\) M1
Solving quadratic for \(\tan\theta\) oe
\(\Rightarrow \tan\theta = 0\) or \(2\) \(\Rightarrow \theta = 0\) or \(63.43\) A1 A1
www. \(-1\) extra solutions in the range.
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## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\tan(\theta+45) = \frac{\tan\theta + \tan 45}{1 - \tan\theta\tan 45} = \frac{\tan\theta + 1}{1 - \tan\theta}$ | M1 A1 | oe using sin/cos |
| $\Rightarrow \frac{\tan\theta + 1}{1-\tan\theta} = 1 - 2\tan\theta$ | | |
| $\Rightarrow 1 + \tan\theta = (1-2\tan\theta)(1-\tan\theta) = 1 - 3\tan\theta + 2\tan^2\theta$ | M1 A1 | Multiplying up and expanding. Any correct one line equation. |
| $\Rightarrow 0 = 2\tan^2\theta - 4\tan\theta = 2\tan\theta(\tan\theta - 2)$ | M1 | Solving quadratic for $\tan\theta$ oe |
| $\Rightarrow \tan\theta = 0$ or $2$ | | |
| $\Rightarrow \theta = 0$ or $63.43$ | A1 A1 | www. $-1$ extra solutions in the range. |
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4 Solve the equation $\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }$.
\hfill \mbox{\textit{OCR MEI C4 Q4 [7]}}