### Instruction: Compute $\cos \left( \arcsin \frac{2}{3} \right).$.### Response: I need to find the cosine of an angle whose sine is 2/3.
I can use a right triangle to visualize this situation.
If I let the opposite side be 2 and the hypotenuse be 3, then the adjacent side must be $\sqrt{3^2 - 2^2} = \sqrt{5}$ by the Pythagorean theorem.
Now I have a right triangle with sides 2, $\sqrt{5}$, and 3.
The angle whose sine is 2/3 is the angle opposite the side of length 2.
The cosine of that angle is the ratio of the adjacent side to the hypotenuse.
So, $\cos \left( \arcsin \frac{2}{3} \right) = \frac{\sqrt{5}}{3}$.