Question

Question #1dd55

Answer 1

`x=30+360*N` and
`x=150+360*N`
where `N` is any integer number (positive, negative or zero).

Explanation:

Looks like numeric values 30, and 60 in the original equation are degrees. So, all the numbers occurred as arguments in trigonometric functions in this explanation are degrees as well.

Let's temporarily substitute the value of `sin(x)` as `y`.
Also, let's take into consideration that
`cos(60) = 1/2`
`cos(30) = sqrt(3)/2`
`sin(60) = sqrt(3)/2`

With this in mind, our equation looks like this:
`2y = y*1/2 + sqrt(3)/2*sqrt(3)/2`

Multiplying left and right parts by `4` and taking into account that `sqrt(3)*sqrt(3) = 3`, we get the following equation:
`8y = 2y+3` or
`6y = 3`

The solution to the above equation is `y=1/2`.

Now recall that `y` was a substitution for `sin(x)`. So, we have the equation for `x`:
`sin(x) = 1/2`

First of all, we know that `sin(30)=1/2`. Therefore, `x=30` is the solution. But it's not the only one.

There are multiple solutions to this trigonometric equation based on the following properties of function `sin(x)`:
`sin(x) = sin(x+360)`
`sin(x) = sin(180-x)`

Therefore, from one "main" value `x=30` we derive a set of other values:
`x=30+360*N` and
`x=150+360*N`
where `N` is any integer number (positive, negative or zero).