Answer
No, there does not exist an angle with the function values $\cos$$\theta$ = $\frac{2}{3}$ and $\sin$$\theta$ = $\frac{3}{4}$.
Work Step by Step
Using the Trigonometric Identity $\sin$$^{2}$ $\theta$ + $\cos$$^{2}$ $\theta$ = 1 we can determine if this is true.
$\sin$$^{2}$ $\theta$ + $\cos$$^{2}$ $\theta$
= ($\frac{3}{4}$)$^{2}$ + ($\frac{2}{3}$)$^{2}$
= ($\frac{9}{16}$) + ($\frac{4}{9}$)
= $\frac{145}{144}$
$\frac{145}{144}$ $\ne$ 1
Since the trigonometric identity has been not been satisfied the angle does not exist.
