Question

Solve in R Sin (x/4) = √2/2

Answer 1

Trigonometric Equations

Solve for x in R:

$sin\text{ }\frac{x}{4}=\frac{\sqrt{2}}{2}$

There are two angles whose sine is the given value: 45° and 135°. We need to express them in radians:

$45^o=45\text{ }\frac{\pi}{180}=\frac{\pi}{4}$$135^o=135\frac{\pi}{180}=\frac{3\pi}{4}$

Thus, we have two solutions:

$\begin{gathered} \frac{x}{4}=\frac{\pi}{4} \\ x=\pi \\ And: \\ \frac{x}{4}=\frac{3\pi}{4} \\ x=3\pi \end{gathered}$

Both solutions point to the same terminal angle, so we only have one solution in the first rotation of the angle:

x = π

Since it's required to find the solution for all the real numbers, we must account for all the possible angles in any number of rotations clockwise or counterclockwise as follows:

x = π + 2kπ

Where k is an integer number. For example, for k = 1, we have the already-found solution above x = 3π