First slide

Trigonometric equations

Question

General solution of $\tan θ + \tan 4 θ + \tan 7 θ = \tan θtan 4 θ \tan 7 θ$ is

Moderate

A

$θ = \frac{nπ}{12},$ where $n \in Z$
B

$θ = \frac{nπ}{9},$ where $n \in Z$
C

$θ = nπ + \frac{π}{12},$ where $n \in Z$
D

none of these

Solution

From the given equation, we have
$\begin{array}{l} or \frac{\tan θ + \tan 4 θ}{1 - \tan θtan 4 θ} = - \tan 7 θ \\ or \tan (θ + 4 θ) = - \tan 7 θ \\ or \tan 5 θ = \tan (- 7 θ) \\ \Rightarrow 5 θ = nπ - 7 θ \\ or θ = nπ / 12, where n \in Z, but n \neq 6, 18, 30, \dots \end{array}$

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