To determine the value of \(\(\tan 30^\circ\)\), we can use trigonometric principles from right-angled triangles and the unit circle.
Here's a detailed step-by-step process:
1. Understanding the Trigonometric Function:
- The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, \(\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)\).
2. Special Right Triangle:
- One common way to derive \(\(\tan 30^\circ\)\) is by using a 30-60-90 right triangle. In such a triangle, the angles are 30°, 60°, and 90°. The sides of this triangle have a known ratio:
- The side opposite the 30° angle (shorter leg) is \(\( \frac{1}{2} \)\) times the hypotenuse.
- The side opposite the 60° angle (longer leg) is \(\( \frac{\sqrt{3}}{2} \)\) times the hypotenuse.
- The hypotenuse is considered to be 1 unit in length for simplicity.
3. Calculating \(\(\tan 30^\circ\)\):
- In a 30-60-90 triangle where the hypotenuse is 1 unit:
- The length of the side opposite the 30° angle (shorter leg) is \(\( \frac{1}{2} \)\).
- The length of the side adjacent to the 30° angle (longer leg) is \(\( \frac{\sqrt{3}}{2} \)\).
- Therefore, \(\(\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \)\).
4. Simplifying the Result:
- \(\(\frac{1}{\sqrt{3}}\)\) can be rationalized to \(\(\frac{\sqrt{3}}{3}\)\). However, in decimal form, this value is approximately \(\(0.5773502691896257\)\).
Thus, the value of \(\(\tan 30^\circ\)\) is \(\(0.5773502691896257\)\).
