First slide

Trigonometric equations

Question

The general solution of the equation $2 \cot \frac{θ}{2} = {(1 + \cot θ)}^{2}$ is

Moderate

A

$n π + {(- 1)}^{n} \frac{π}{4}, n \in Z$
B

$n π + {(- 1)}^{n} \frac{π}{3}, n \in Z$
C

$n π + {(- 1)}^{n} \frac{π}{6}, n \in Z$
D

$\frac{n π}{2}$

Solution

We have $2 \cot \frac{θ}{2} = {(1 + \cot θ)}^{2}$
$\Rightarrow \frac{2 \times 2 \cos \frac{θ}{2} \times \cos \frac{θ}{2}}{2 \sin \frac{θ}{2} \times \cos \frac{θ}{2}} = {(1 + \cot θ)}^{2}$
$\Rightarrow \frac{2 (1 + \cos θ)}{\sin θ} = \cos e c^{2} θ + 2 \cot θ$
$\Rightarrow 2 \cos e c θ + 2 co t θ = \cos e c^{2} θ + 2 \cot θ$
$\Rightarrow \cos e c θ = 2$
$\Rightarrow \sin θ = \frac{1}{2}$
$\Rightarrow θ = n π + {(- 1)}^{n} \frac{π}{6}, n \in Z$

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