Question 251209
2sin(x)=sin(120-x)<br>
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.<br>
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(3)/2}}} and cos(120) = -1/2:
{{{2sin(x) = (sqrt(3)/2)cos(x) - (-1/2)sin(x)}}}
or
{{{2sin(x) = (sqrt(3)/2)cos(x) + (1/2)sin(x)}}}
Subtracting {{{(1/2)}}}sin(x) from each side we get:
{{{(3/2)sin(x) = (sqrt(3)/2)cos(x)}}}
Multiply both sides by 2 to get rid of the fractions:
{{{3sin(x) = (sqrt(3))cos(x)}}}
Divide both sides by cos(x):
{{{(3sin(x))/cos(x) = sqrt(3)}}}
Since sin(x)/cos(x) = tan(x):
{{{3tan(x) = sqrt(3)}}}
Divide both sides by 3:
{{{tan(x) = sqrt(3)/3}}}
We now have the desired form. This should be a recognizable this value for tan. (If not, use your calculator.)
x = 30 + 180n