Solution:
Consider the following information:
[tex]\cos (\theta)=\frac{2}{3}[/tex]and
[tex]\tan (\theta)<0[/tex]by definition, this means that:
[tex]\cos (\theta)=\frac{2}{3}=\frac{adjacent\text{ side}}{hypotenuse}[/tex]now, to find sin(θ), we must first apply the Pythagorean theorem to find the opposite side:
[tex]\text{opposite side = }\sqrt[]{3^2-2^2}\text{ = }\sqrt[]{5}[/tex]now, since tan θ < 0 and cos θ > 0, then, the angle must be in the second quadrant, thus sin θ < 0 and it is:
[tex]\sin (\theta)=-\frac{opposite\text{ side}}{hypotenuse}=-\frac{\sqrt[]{5}}{3}[/tex]So that, we can conclude that the correct answer is:
[tex]-\frac{\sqrt[]{5}}{3}[/tex]
