1997 Paper 2 Q6

Year: 1997
Paper: 2
Question Number: 6

Course: LFM Pure
Section: Trigonometry 2

Difficulty: 1600.0 Banger: 1500.0

trigonometry tan identity double angle formula triple angle formula trigonometric equations cot quadratic in tan

Problem

Show that, if $\,\tan^2\phi=2\tan\phi+1$, then $\tan2\phi=-1$. Find all solutions of the equation $$\tan\theta=2+\tan3\theta$$ which satisfy $0<\theta< 2\pi$, expressing your answers as rational multiples of $\pi$. Find all solutions of the equation the equation $$\cot\theta=2+\cot3\theta$$ which satisfy $$-\frac{3\pi}{2}<\theta<\frac{\pi}{2}.$$

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Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1500.0

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Problem source

Show that, if $\,\tan^2\phi=2\tan\phi+1$, then $\tan2\phi=-1$.
Find all solutions of the equation
$$\tan\theta=2+\tan3\theta$$
which satisfy $0<\theta< 2\pi$,
expressing your answers as rational multiples of $\pi$. 
Find all solutions of the equation
the equation
$$\cot\theta=2+\cot3\theta$$
which satisfy $$-\frac{3\pi}{2}<\theta<\frac{\pi}{2}.$$

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