if sin theta = 2/3 and tan theta <0 what is the value of cos theta?a) (sqrt5)/2b) -sqrt5c) (sqrt5)/3d) -(sqrt5)/3

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]