Respuesta :
Answer:
3 solutions:
[tex]\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}} [/tex]
Step-by-step explanation:
So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is [tex]tan(\frac{\theta}{2})[/tex] so let's focus on that part of the equation first.
We know that:
[tex]tan(\frac{\theta}{2})=\frac{sin(\frac{\theta}{2})}{cos(\frac{\theta}{2})}[/tex]
therefore:
[tex]cos(\frac{\theta}{2})\neq0[/tex]
so we need to find the angles that will make the cos function equal to zero. So we get:
[tex]cos(\frac{\theta}{2})=0[/tex]
[tex]\frac{\theta}{2}=cos^{-1}(0)[/tex]
[tex]\frac{\theta}{2}=\frac{\pi}{2}+\pi n[/tex]
or
[tex] \theta=\pi+2\pi n[/tex]
we can now start plugging values in for n:
[tex] \theta=\pi+2\pi (0)=\pi[/tex]
if we plugged any value greater than 0, we would end up with an angle that is greater than [tex]2\pi[/tex] so, that's the only angle we cannot include in our answer set, so:
[tex]\theta\neq \pi[/tex]
having said this, we can now start solving the equation:
[tex]tan(\frac{\theta}{2})=sin(\theta)[/tex]
we can start solving this equation by using the half angle formula, such a formula tells us the following:
[tex]tan(\frac{\theta}{2})=\frac{1-cos(\theta)}{sin(\theta)}[/tex]
so we can substitute it into our equation:
[tex]\frac{1-cos(\theta)}{sin(\theta)}=sin(\theta)[/tex]
we can now multiply both sides of the equation by [tex]sin(\theta)[/tex]
so we get:
[tex]1-cos(\theta)=sin^{2}(\theta)[/tex]
we can use the pythagorean identity to rewrite [tex]sin^{2}(\theta)[/tex] in terms of cos:
[tex]sin^{2}(\theta)=1-cos^{2}(\theta)[/tex]
so we get:
[tex]1-cos(\theta)=1-cos^{2}(\theta)[/tex]
we can subtract a 1 from both sides of the equation so we end up with:
[tex]-cos(\theta)=-cos^{2}(\theta)[/tex]
and we can now add [tex]cos^{2}(\theta)[/tex]
to both sides of the equation so we get:
[tex]cos^{2}(\theta)-cos(\theta)=0[/tex]
and we can solve this equation by factoring. We can factor [tex]cos(\theta)[/tex] to get:
[tex]cos(\theta)(cos(\theta)-1)=0[/tex]
and we can use the zero product property to solve this, so we get two equations:
Equation 1:
[tex]cos(\theta)=0[/tex]
[tex]\theta=cos^{-1}(0)[/tex]
[tex]\theta={\frac{\pi}{2}, \frac{3\pi}{2}}[/tex]
Equation 2:
[tex]cos(\theta)-1=0[/tex]
we add a 1 to both sides of the equation so we get:
[tex]cos(\theta)=1[/tex]
[tex]\theta=cos^{-1}(1)[/tex]
[tex]\theta=0[/tex]
so we end up with three answers to this equation:
[tex]\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}} [/tex]
