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.
I would gather terms, solve for cos x/2, then solve for x/2, then solve for x. You will have more than one solution
cosx = 1 - 2sin^2 x/2
Now you have a quadratic in u=sin x/2
u = 1-(1-2u^2)
To solve the equation sin(x/2) = 1 - cos(x), you are correct that you can use the half-angle identity:
sin(x/2) = ±√(1 - cos(x))/2.
Here's how you can proceed to solve the equation:
1. Square both sides of the equation to remove the square root:
(sin(x/2))² = (1 - cos(x))/2.
2. Expand the left side of the equation using the double-angle identity for sine:
(1 - cos(x))/2 = (1 - cos(x))/2 [(1 - cos(x))/2].
Simplify the equation:
(1 - cos(x))/2 = (1 - 2cos(x) + cos²(x))/4.
3. Multiply both sides of the equation by 4 to eliminate the denominator:
2(1 - cos(x)) = 1 - 2cos(x) + cos²(x).
4. Expand the left side of the equation:
2 - 2cos(x) = 1 - 2cos(x) + cos²(x).
5. Rearrange the terms:
0 = cos²(x) - cos(x) + 1.
6. Recognize that this is a quadratic equation in terms of cos(x). Let's let t = cos(x). The equation becomes:
0 = t² - t + 1.
7. Solve this quadratic equation. However, this quadratic equation has no real solutions because its discriminant (b² - 4ac) is negative, meaning it does not intersect the x-axis.
Therefore, the original equation sin(x/2) = 1 - cos(x) has no real solutions.
Note: While the half-angle identity sin(x/2) = ±√(1 - cos(x))/2 is useful in many cases, it might not always be relevant or provide real solutions. It is always important to check the validity and limitations of the given identities when solving equations.