Question 372386: Can someone please explain how to find all solutions of sin(x/2)+cosx-1=0? I have been out of school for quite some time and I don't remember how to do this.
Found 2 solutions by robertb, jsmallt9:
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! From double -angle identity,  . By direct substitution,
 . Hence
 , or  . (The 2nd is the same as  .)
From the 1st equation we get x = 0 degrees, or 360 degrees. From the second, we get x = 60 degrees, or 300 degrees.
****The solution given by the author below says that I have given ONLY TWO solutions, when I have given FOUR, namely 0, 60, 300, or 360 degrees. If he had read my ENTIRE solution he should HAVE REALIZED that I was going for solutions between 0 and 360 degrees, inclusive. I have left it to asker to just find the OTHER INFINITE solutions for his consideration.****
Answer by jsmallt9(3758)  (Show Source):
You can put this solution on YOUR website! The solution provided by another tutor is not quite complete. Only two angles are given as solutions where there should be an infinite set of angles.
He/she is correct up to
sin(x/2) = 0 or sin(x/2) = 1/2
From here we have to ask ourselves...- Where is sin = 0? Answer: At 0 and 180 degrees and all angles that are coterminal with these.
- Where is sin = 1/2? Answer: At 30 and 150 degrees and all angles that are coterminal with these.
So
x/2 = 0 + 360n (where n is any integer)
x/2 = 180 + 360n (where n is any integer)
x/2 = 30 + 360n (where n is any integer)
x/2 = 150 + 360n (where n is any integer)
(The "+ 360n (where n is any integer)" is how we express "and all the coterminal angles")
All we do now is multiply each side of all four equations by 2:
x = 0 + 720n (where n is any integer)
x = 360 + 720n (where n is any integer)
x = 60 + 720n (where n is any integer)
x = 300 + 720n (where n is any integer)
These four equations describe all the angles which are solutions to your equation.
BTW: The first two equations can be "condensed" to a single equation:
x = 0 + 360n (where n is any integer)
See if you understand why this equation generates all the same angles as
x = 0 + 720n (where n is any integer)
x = 360 + 720n (where n is any integer)
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