To determine the value of [tex]\(\tan 60^{\circ}\)[/tex], let's proceed with a detailed explanation using trigonometric principles.
### Step-by-Step Solution
1. Understanding the Angle and the Unit Circle:
[tex]\(\tan 60^{\circ}\)[/tex] is the tangent of a 60-degree angle. The tangent function ([tex]\(\tan \theta\)[/tex]) for an angle [tex]\(\theta\)[/tex] in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
2. Recognizing the 30-60-90 Triangle:
A 30-60-90 triangle is a special right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. The side lengths of a 30-60-90 triangle have a specific ratio:
- The side opposite the 30-degree angle is the shortest and is often designated as [tex]\(x\)[/tex].
- The side opposite the 60-degree angle is [tex]\(x\sqrt{3}\)[/tex].
- The hypotenuse is [tex]\(2x\)[/tex].
3. Calculating the Tangent:
For [tex]\(\tan 60^{\circ}\)[/tex], we use the 30-60-90 triangle. For this triangle:
[tex]\[
\tan 60^{\circ} = \frac{\text{opposite}}{\text{adjacent}} = \frac{x\sqrt{3}}{x} = \sqrt{3}
\][/tex]
4. Verifying the Correct Answer:
From the above calculation, the value of [tex]\(\tan 60^{\circ}\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
### Conclusion
The correct option for [tex]\(\tan 60^{\circ}\)[/tex] is [tex]\( A. \sqrt{3} \)[/tex].
Hence, the answer is:
A. [tex]\(\sqrt{3}\)[/tex]
