First slide

Trigonometric equations

Question

The general solution of the trigonometrical equation $\sin x + \cos x = 1$ is

Moderate

A

$x = 2 n π, n \in I$
B

$x = 2 n π + \frac{π}{2}, n \in I$
C

$x = n π + (- 1)^{n} \frac{π}{4} - \frac{π}{4}, n \in I$
D

none of these

Solution

$\sin x + \cos x = 1 \Rightarrow \sin (x + π / 4) = \frac{1}{\sqrt{2}}$
$\begin{array}{l} = \sin \frac{π}{4} \\ \Rightarrow x + \frac{π}{4} = n π + (- 1)^{n} \frac{π}{4} \\ \Rightarrow x = n π + (- 1)^{n} \frac{π}{4} - \frac{π}{4} \end{array}$
which includes $x = 2 n π$ and $x = 2 n π + \frac{π}{2}$ .

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