we have the following
[tex]\sin \theta=\frac{4}{9}[/tex]
sin of an angle is the same as:
[tex]\sin \theta=\frac{opposite}{hypotenuse}[/tex]
therefore we can create the following right triangle:
we can calculate the adjacent side using the pythagorean theorem
[tex]h^2=a^2+b^2[/tex]
where h is the hypotenuse, a is the adjacent side and b the opposite side to the angle.
thus, the adjacent side is:
[tex]a=\sqrt[]{h^2-b^2}=\sqrt[]{9^2-4^2}=\sqrt[]{81-16}=\sqrt[]{65}[/tex]
Using that value, we can now calculate cos of the angle
[tex]\cos \theta=\frac{adjacent}{hypotenuse}[/tex][tex]\cos \theta=\frac{\sqrt[]{65}}{9}[/tex]
which can't be simplify, thus that is the answer for the exact value
