First slide

Trigonometric equations

Question

The number of solutions of $\sec x - \tan x = \sqrt{3}, x \in [0, 3 π]$ is

Easy

A

4
B

3
C

2
D

1

Solution

We have $\sec x - \tan x = \sqrt{3}$
$\Rightarrow \sec x + \tan x = \frac{1}{\sqrt{3}}$
Adding $2 \sec x = \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}}$
$\Rightarrow \sec x = \frac{2}{\sqrt{3}} > 0$
Subtracting, we get $2 \tan x = \frac{1}{\sqrt{3}} - \sqrt{3} = \frac{- 2}{\sqrt{3}}$
$\Rightarrow \tan x = \frac{- 1}{\sqrt{3}} < 0$
$\Rightarrow x b e l o n g s t o 4 t h q u a d r a n t$
$\Rightarrow x = \frac{11 π}{6}, \frac{23 π}{6}$

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