Answer:
[tex]\cos(\theta)=-\frac{\sqrt{3}}{2}[/tex]
Step-by-step explanation:
If [tex]\theta[/tex] is between [tex]\pi[/tex] and
[tex]\frac{3\pi}{2}[/tex] , then we are in third quadrant. This is where x (cosine value) and y (sine value) are both negative.
We are going to use a Pythagorean Identity, namely [tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex].
We are going to plug -1/2 for [tex]\sin(\theta)[/tex].
That gives us this equation to solve for [tex]\cos(\theta)[/tex]:
[tex](\frac{-1}{2})^2+\cos^2(\theta)=1[/tex]
[tex]\frac{1}{4}+\cos^2(\theta)=1[/tex]
Subtract 1/4 on both sides:
[tex]\cos^2(\theta)=\frac{3}{4}[/tex]
Square root both sides:
[tex]\cos(\theta)=\sqrt{\frac{3}{4}}[/tex]
[tex]\cos(\theta)=\pm \frac{\sqrt{3}}{2}[/tex]
We already determine cosine value was negative.
[tex]\cos(\theta)=-\frac{\sqrt{3}}{2}[/tex]
