How do we use cosine rule in trigonometry

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

$$
c^2 = (5)^2 + (7)^2 - 2(5)(7) \cos 60^\circ
$$

$$
c^2 = 25 + 49 - 70 \times 0.5
$$
$$
c^2 = 74 - 35
$$
$$
c^2 = 39
$$

$$
c = \sqrt{39}
$$
$$
c \approx 6.245
$$
The length of side $c$ is approximately $\boxed{6.25 \, \text{cm}}$ (to 3 significant figures).

**Example:**  
Consider a triangle with sides $a = 3 \, \text{cm}$, $b = 4 \, \text{cm}$, and $c = 5 \, \text{cm}$. Find the measure of angle $C$.

Starting from $c^2 = a^2 + b^2 - 2ab \cos C$:
$$
2ab \cos C = a^2 + b^2 - c^2
$$
$$
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
$$

$$
\cos C = \frac{(3)^2 + (4)^2 - (5)^2}{2(3)(4)}
$$

$$
\cos C = \frac{9 + 16 - 25}{24}
$$
$$
\cos C = \frac{25 - 25}{24}
$$
$$
\cos C = \frac{0}{24}
$$
$$
\cos C = 0
$$

$$
C = \arccos(0)
$$
$$
C = 90^\circ
$$
The measure of angle $C$ is $\boxed{90^\circ}$. (This particular example forms a right-angled triangle, which is a special case).