Question

Calculate x? Sin(x+60)=2Sinx

Answer 1

`x=pi/3+2kpi`

Explanation:

We have
`sin(x+pi/3)=sin(x)cos(pi/3)+cos(x)sin(pi/3)=2sin(x)`
Dividing by `sin(x)`
`cos(pi/3)+cot(x)sin(pi/3)=2`
`cot(x)=(2-cos(pi/3))/sin(pi/3)`
so

`tan(x)=sin(pi/3)/(2-cos(pi/3))=1/sqrt(3)`

Answer 2

`x = 30 + 360n`

Explanation:

First, we apply compound angle formula on `sin(x+60)`.

`sin(x+60) = sin(x)cos(60) + sin(60)cos(x) = 1/2sin(x) + sqrt(3)/2cos(x)`

We now have:
`2sin(x) = 1/2sin(x) + sqrt(3)/2cos(x)`

Since `sin(x)` is not equal to 0 (if `sin(x)` is equal to 0, it is not possible for `sin(x+60)` to be equal to 0 as well), we can divide both sides of the equation by `sin(x)`.

`2 = 1/2 + sqrt(3)/(2tan(x))`

Making `tan(x)` the subject,

`3/2 = sqrt(3)/(2tan(x))`
`tan(x) = 1/sqrt(3)`.

Therefore,

`x = 30 + 360n`

The `360n` is because trigonometric functions are periodic about 360 degrees, or 2`pi` radians, which means the equation will still hold no matter how much you add or subtract 360 degrees from x.