SOLUTION: Hi. I am trying to solve for x in the following equation: 2sin(x)=sin(120-x) Your help would be greatly appreciated. Thanks. -Max

Algebra ->  Trigonometry-basics -> SOLUTION: Hi. I am trying to solve for x in the following equation: 2sin(x)=sin(120-x) Your help would be greatly appreciated. Thanks. -Max      Log On


   

Question 251209: Hi.
I am trying to solve for x in the following equation:
2sin(x)=sin(120-x)
Your help would be greatly appreciated. Thanks.
-Max
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2sin(x)=sin(120-x)

Generally we can solve an equation if we can solve a Trig. equation if we can transform it into the form:
q(variable-expression) = number
where q is one of the Trig functions.

We have two sin's but with different arguments. SO we cannot combine them into a single function as they stand. So we need to use one of the identities to change the argument. We will use sin(A-B) = sin(A)cos(B) - cos(A)sin(B) giving us:
2sin(x) = sin(120)cos(x) - cos(120)sin(x)
Since sin(120) = sqrt%283%29%2F2 and cos(120) = -1/2:
2sin%28x%29+=+%28sqrt%283%29%2F2%29cos%28x%29+-+%28-1%2F2%29sin%28x%29
or
2sin%28x%29+=+%28sqrt%283%29%2F2%29cos%28x%29+%2B+%281%2F2%29sin%28x%29
Subtracting %281%2F2%29sin(x) from each side we get:
%283%2F2%29sin%28x%29+=+%28sqrt%283%29%2F2%29cos%28x%29
Multiply both sides by 2 to get rid of the fractions:
3sin%28x%29+=+%28sqrt%283%29%29cos%28x%29
Divide both sides by cos(x):
%283sin%28x%29%29%2Fcos%28x%29+=+sqrt%283%29
Since sin(x)/cos(x) = tan(x):
3tan%28x%29+=+sqrt%283%29
Divide both sides by 3:
tan%28x%29+=+sqrt%283%29%2F3
We now have the desired form. This should be a recognizable this value for tan. (If not, use your calculator.)
x = 30 + 180n