Question 458991
Solve for x in the interval (0,360), sin(1/2)x = sin(x)
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{{{sin(x/2) = sin(x)}}}
We can write this as
{{{sin(x/2) - sin(x) = 0}}}
Let {{{w = x/2}}}
Then the equation becomes
{{{sin(w) - sin(2w) = 0}}}
Using the double-angle formula {{{sin(2w) = 2sin(w)cos(w)}}}, we have:
{{{sin(w) - 2sin(w)cos(w) = 0}}}
Pull out {{{sin(w)}}}:
{{{sin(w)(1 - 2 cos(w)) = 0}}}
The RHS will equal 0 if {{{sin(w) = 0}}} or {{{1 - 2cos(w) = 0}}}
There is no solution for {{{sin(w) = 0}}} on the open interval between 0 and 360 deg.
So we need only to solve {{{1 - 2cos(w) = 0}}}
This gives {{{cos(w) = 1/2}}}
So w = 60 deg.
This means x = 2w = 120 deg.
Ans: x = 120 deg.