How to solve sin(x/2) = 1 - cos x? I know you would use the half angle identity sin(x/2) = +-(√1 - cos x/2), just don't know what to do from there.
To solve the equation sin(x/2) = 1 - cos(x), you can indeed start by using the half angle identity for sine:
sin(x/2) = ±√(1 - cos(x))/2
Let's take the positive square root for now:
sin(x/2) = √(1 - cos(x))/2
To proceed, we can square both sides of the equation to eliminate the square root:
(sin(x/2))^2 = (√(1 - cos(x))/2)^2
Simplifying further:
sin^2(x/2) = (1 - cos(x))/4
Now, let's use the identity sin^2(x/2) = (1 - cos(x))/2:
(1 - cos(x))/2 = (1 - cos(x))/4
To continue, we can cross-multiply:
2(1 - cos(x)) = 1 - cos(x)
Expanding and rearranging the equation:
2 - 2cos(x) = 1 - cos(x)
2cos(x) - cos(x) = 1 - 2
Combining like terms:
cos(x) = -1
Now we have found the value of cos(x). To find the value of x, we need to consider the range of the cosine function, which is [-1, 1].
In this case, cos(x) is equal to -1, which implies that x is an odd multiple of π:
x = (2n + 1)π, where n is an integer.
Therefore, the solution to the equation sin(x/2) = 1 - cos(x) is x = (2n + 1)π, where n is an integer.