4. Find the exact value of $b$ in a triangle $A B C$ with the given information. $A=45^{\circ}, B=60^{\circ}, a=10$

Step 1 ：We are given a triangle ABC with angles A = 45 degrees, B = 60 degrees, and side a = 10. We are asked to find the length of side b.

Step 2 ：We can use the Law of Sines to solve this problem. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle.

Step 3 ：Setting up the equation using the Law of Sines, we get \(\frac{a}{\sin A} = \frac{b}{\sin B}\).

Step 4 ：Substituting the given values into the equation, we get \(\frac{10}{\sin 45^\circ} = \frac{b}{\sin 60^\circ}\).

Step 5 ：Solving for b, we get \(b = \frac{10 \times \sin 60^\circ}{\sin 45^\circ}\).

Step 6 ：Calculating the above expression, we find that b is approximately 12.24744871391589.

Step 7 ：Final Answer: The exact value of b in the triangle ABC is \(\boxed{12.24744871391589}\).