First slide

Trigonometric equations

Question

The total number of solutions of $\cos x = \sqrt{1 - \sin 2 x}$ in $[0, 2 π]$ is equal to

Moderate

A

2
B

3
C

5
D

none of these

Solution

$\cos x = \sqrt{1 - \sin 2 x} = | \sin x - \cos x |$
$\begin{array}{l} (a) \sin x < \cos x \\ \Rightarrow x \in [0, \frac{π}{4}) \cup (\frac{5 π}{4}, 2 π] \end{array}$
Then the given equation is
$\begin{array}{l} \cos x = \cos x - \sin x \\ or \sin x = 0 \\ \Rightarrow x = 2 π \end{array}$ [from Eq. (i)]
$\begin{array}{l} (b) \sin x \geq \cos x \\ \Rightarrow x \in (\frac{π}{4}, \frac{5 π}{4}) \\ \Rightarrow \cos x = \sin x - \cos x \\ or \tan x = 2 \\ \Rightarrow x = \tan^{- 1} 2 \end{array}$
Hence, there are two solutions.

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