First slide

Trigonometric equations

Question

The smallest positive value of x (in radians) satisfying the equation $\log_{\cos x} (\frac{\sqrt{3}}{2} \sin x) = 2 - \log_{\sec x} (\tan x)$ , is

NA

A

$\frac{π}{12}$
B

$\frac{π}{6}$
C

$\frac{π}{4}$
D

$\frac{π}{3}$

Solution

We have $\log_{\cos x} (\frac{\sqrt{3}}{2} \sin x) - \log_{\cos x} (\tan x) = 2$
$\begin{array}{l} \Rightarrow \log_{\cos x} (\frac{\sqrt{3}}{2} \cos x) = 2 \\ \Rightarrow \frac{\sqrt{3}}{2} \cos x = \cos^{2} x \\ \Rightarrow \cos x (\cos x - \frac{\sqrt{3}}{2}) = 0 \\ ∴ x = \frac{π}{6} \\ (x = \frac{π}{2} is rejected) \end{array}$

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