Question 372386
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.<br>
He/she is correct up to
sin(x/2) = 0 or sin(x/2) = 1/2
From here we have to ask ourselves...<ul><li>Where is sin = 0? Answer: At 0 and 180 degrees <i>and all angles that are coterminal with these.</i></li><li>Where is sin = 1/2? Answer: At 30 and 150 degrees <i>and all angles that are coterminal with these.</i></li></ul>
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")<br>
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 <i>all</i> the angles which are solutions to your equation.<br>
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)