Suppose $A B C$ is a right triangle with sides $a, b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find the values of the six trigonometric functions for angle $B$. Rationalize the denominators when applicable. \[ a=3, b=4 \] What is the length of side $c$ ? \[ c=5 \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \sin B=\frac{4}{5} \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \cos \mathrm{B}= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Step 1 ：Given a right triangle $ABC$ with sides $a=3$, $b=4$, and $c$ with right angle at $C$.

Step 2 ：Use the Pythagorean theorem $a^2 + b^2 = c^2$ to find the length of side $c$. Substituting the given values, we get $3^2 + 4^2 = c^2$, which simplifies to $9 + 16 = c^2$ and further simplifies to $25 = c^2$. Taking the square root of both sides, we find that $c = \sqrt{25} = 5$.

Step 3 ：Next, find the values of the six trigonometric functions for angle $B$. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the sine of angle $B$ is equal to $b/c = 4/5$.

Step 4 ：The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In this case, the cosine of angle $B$ is equal to $a/c = 3/5$.

Step 5 ：Final Answer: The length of side $c$ is \(\boxed{5}\) and $\cos B = \boxed{\frac{3}{5}}$.