To determine the value of [tex]\(\tan 30^\circ\)[/tex], 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, [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex].
2. Special Right Triangle:
- One common way to derive [tex]\(\tan 30^\circ\)[/tex] 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 [tex]\( \frac{1}{2} \)[/tex] times the hypotenuse.
- The side opposite the 60° angle (longer leg) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] times the hypotenuse.
- The hypotenuse is considered to be 1 unit in length for simplicity.
3. Calculating [tex]\(\tan 30^\circ\)[/tex]:
- In a 30-60-90 triangle where the hypotenuse is 1 unit:
- The length of the side opposite the 30° angle (shorter leg) is [tex]\( \frac{1}{2} \)[/tex].
- The length of the side adjacent to the 30° angle (longer leg) is [tex]\( \frac{\sqrt{3}}{2} \)[/tex].
- Therefore, [tex]\(\tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \)[/tex].
4. Simplifying the Result:
- [tex]\(\frac{1}{\sqrt{3}}\)[/tex] can be rationalized to [tex]\(\frac{\sqrt{3}}{3}\)[/tex]. However, in decimal form, this value is approximately [tex]\(0.5773502691896257\)[/tex].
Thus, the value of [tex]\(\tan 30^\circ\)[/tex] is [tex]\(0.5773502691896257\)[/tex].
