To find the length of side $c$ in a triangle where the sides $a = 2$, $b = 6$, and the angle between them $C = 40^\circ$, we can use the Law of Cosines:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Substituting the known values:

$c^2 = 2^2 + 6^2 - 2(2)(6) \cdot \cos(40^\circ)$

$c^2 = 4 + 36 - 24 \cdot \cos(40^\circ)$

Let me calculate the value of $c$.The length of side $c$ is approximately $4.65$ units.

Would you like more details or have any other questions?

Here are 5 related questions you might find useful:

1. How does the Law of Cosines apply to different types of triangles?
2. What happens when the angle $C$ is $90^\circ$ in the Law of Cosines?
3. Can you solve for other angles or sides using the Law of Cosines in this triangle?
4. How does the Law of Sines differ from the Law of Cosines in solving triangles?
5. How can you determine if a triangle is obtuse or acute using the Law of Cosines?

Tip: Always ensure the angle is in degrees or radians based on your calculator or tool before applying trigonometric functions.