Answer:
d) [tex]\cos(\theta)=-\frac{\sqrt{5}}{3}[/tex]
Step-by-step explanation:
If [tex]\sin(\theta)=\frac{2}{3}[/tex] and [tex]\tan(\theta)\:<\:0[/tex], then
[tex]\theta[/tex] is in quadrant 2.
Recall that;
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
We substitute the given sine ratio to obtain;
[tex](\frac{2}{3})^2+\cos^2(\theta)=1[/tex]
[tex]\frac{4}{9}+\cos^2(\theta)=1[/tex]
[tex]\cos^2(\theta)=1-\frac{4}{9}[/tex]
[tex]\cos^2(\theta)=\frac{5}{9}[/tex]
[tex]\cos(\theta)=\pm \sqrt{\frac{5}{9}}[/tex]
[tex]\cos(\theta)=\pm \frac{\sqrt{5}}{3}[/tex]
We are in the second quadrant, therefore
[tex]\cos(\theta)=-\frac{\sqrt{5}}{3}[/tex]
