Question

Solve in R Sin (x/4) = √2/2

Answer 1

Trigonometric Equations

Solve for x in R:

`sin\text{ }(x)/(4)=(√(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{ }(\pi)/(180)=(\pi)/(4)`
`135^o=135(\pi)/(180)=(3\pi)/(4)`

Thus, we have two solutions:

`\begin{gathered} (x)/(4)=(\pi)/(4) \\ x=\pi \\ And: \\ (x)/(4)=(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