If the length of each side of a rhombus is 8 cm and its one angle is 60°, then find the lengths of the diagonals of the rhombus.

We know that the diagonals of a rhombus bisect the opposite angles and are perpendicular to each other.

∴ ∠OAB = $\dfrac{60°}{2}$ = 30°.

In right ∠AOB,

⇒ sin 30° = $\dfrac{\text{Perpendicular}}{\text{Hypotenuse}} = \dfrac{OB}{AB}$

⇒ $\dfrac{1}{2} = \dfrac{OB}{AB}$

⇒ OB = $\dfrac{AB}{2}$

⇒ OB = $\dfrac{8}{2}$ = 4 cm.

As diagonals of rhombus bisect each other.

∴ BD = 2 OB = 2 × 4 = 8 cm.

cos 30° = $\dfrac{\text{Base}}{\text{Hypotenuse}} = \dfrac{OA}{AB}$

⇒ $\dfrac{\sqrt{3}}{2} = \dfrac{OA}{AB}$

⇒ OA = $\dfrac{AB\sqrt{3}}{2}$

⇒ OA = $\dfrac{8\sqrt{3}}{2} = 4\sqrt{3}$.

As diagonals of rhombus bisect each other.

∴ AC = 2 OA = $2 \times 4\sqrt{3} = 8\sqrt{3}$.

Hence, the length of the diagonals of the rhombus are 8 cm and $8\sqrt{3}$ cm.