Question

how do I find all the solutions in this equation? csc(θ/2) =sin (θ/2)

Answer 1

The solutions to the equation occur whenever csc(θ/2) =sin (θ/2). In other words, just graph it and observe visually

Answer 2

where both intersect each other

Answer 3

we dont solve the questions graphing them...do you have another method to solve it?

Answer 4

csc (theta/2) = sin (theta/2)
1/sin(theta/2) = sin (theta/2)
1 = sin^2(theta/2)

solve for theta + k2pi where k is an element of an integer

Answer 5

srry 4pi not 2pi

Answer 6

one is the reciprocal of the other, so imagine you were trying to solve \(x=\frac{1}{x}\) what would you get ?

Answer 7

hopefully you get \(x=1\) or \(x=-1\) telling you \(\sin(\frac{\theta}{2})=1\) or \(\sin(\frac{\theta}{2})=-1\)

Answer 8

yeah, I got -1 or 1

Answer 9

on the interval \([0,2\pi)\) the occurs at two places, \(\frac{\theta}{2}=\frac{\pi}{2}\) or \(\frac{\theta}{2}=\frac{3\pi}{2}\)

Answer 10

solving for \(\theta\) we get \(\theta=\pi\) or \(\theta=3\pi\)

Answer 11

if you are looking for some general solution, it is an odd integer times \(\pi\) which you could write at \(\theta=(2k+1)\pi, k\in \mathbb{Z}\)

Answer 12

ok, thanks! I got it right! :)