Find the exact value of \( \tan 30^{\circ} \) \( \tan 30^{\circ}=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

The exact value of \( \tan 30^{\circ} \) is given by the formula: \[ \tan 30^{\circ} = \frac{1}{\sqrt{3}} \] You can also express this in a more simplified fractional form: \[ \tan 30^{\circ} = \frac{\sqrt{3}}{3} \] So, \[ \tan 30^{\circ} = \frac{\sqrt{3}}{3} = \square \] Now, let’s keep the fun going! This value comes from the geometry of a 30-60-90 triangle. In such a triangle, the side opposite the 30-degree angle is half the hypotenuse, while the side opposite the 60-degree angle is \( \sqrt{3}/2 \) of the hypotenuse. Thus, \( \tan 30^{\circ} \) is equal to the opposite over the adjacent, leading to our exact value. If you ever want to impress friends at a party, let them know you can derive the values of all trigonometric functions just by drawing right triangles! It’s a classic way of showcasing why trigonometry can be not just useful but also fun. Plus, a little sketch can go a long way in making complex concepts feel simple and relatable.