First slide

Trigonometric equations

Question

The total number of solutions of $| \cot x | = \cot x + \frac{1}{\sin x},$ $x \in [0, 3 π]$ is equal to

Moderate

A

1
B

2
C

3
D

0

Solution

$\begin{array}{ll} | \cot x | = \cot x + \frac{1}{\sin x} \\ If \cot x > 0 \\ \Rightarrow \cot x = \cot x + \frac{1}{\sin x} = 0 \end{array}$
$\Rightarrow \frac{1}{\sin x} = 0$ ,which is not possible
If $\cot x \leq 0$
$\begin{array}{l} \Rightarrow - \cot x = \cot x + \frac{1}{\sin x} \\ \Rightarrow - 2 \cot x = \frac{1}{\sin x} \\ or \cos x = - \frac{1}{2} \\ \Rightarrow x = \frac{2 π}{3}, \frac{8 π}{3} \end{array}$

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