Final answer:
To solve sin(x) = 2sin(x-30)degrees, we apply the sine addition formula and trigonometric identities. We end up with a system of equations that can be solved algebraically or graphically, seeking angles where sin(x) and cos(x) satisfy the equation.
Step-by-step explanation:
To solve the equation sin(x) = 2sin(x-30)degrees for 0≤x≤360 degrees, we can use trigonometric identities and properties. We can apply the sine addition formula to the right side of the equation, which states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B). In this case, A would be x and B would be 30 degrees.
Applying the identity, we get:
sin(x) = 2[sin(x)cos(30) - cos(x)sin(30)]
We know that cos(30) is √3/2 and sin(30) is 1/2, so we'll substitute those in:
sin(x) = 2[(√3/2)sin(x) - (1/2)cos(x)]
Simplifying, we get:
sin(x) = √3sin(x) - cos(x)
Moving all terms to one side, we get:
0 = sin(x)(√3 - 1) - cos(x). This can be solved by looking for angles x where both sin(x) and cos(x) have known values and satisfy the equation. This is a system of equations and we can use substitution or other algebraic techniques to find x. We might also use graphical methods or numerical solutions. This equation could have multiple solutions over 0 to 360 degrees depending on the unit circle values for sine and cosine.