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

2 Answers
May 27, 2018

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)

May 27, 2018

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 2pi radians, which means the equation will still hold no matter how much you add or subtract 360 degrees from x.