Final answer:
The tangent of 30 degrees is not 3/√3; it is correctly √3/3, as proven using a 30-60-90 right triangle derived from an equilateral triangle with side lengths of 2.
Step-by-step explanation:
The statement provided, "proof of tangent 30 degrees is 3/√3" is incorrect. The accurate value of tangent 30 degrees is √3/3 or 1/√3 after rationalizing the denominator. To prove this, we can look at the properties of an equilateral triangle split into two 30-60-90 right triangles. An equilateral triangle with side lengths of 2 will have a height of √3 by use of the Pythagorean theorem. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle (the height in this case) is √3 times the shorter leg. Therefore, in our example, the shorter leg is 1, the hypotenuse is 2, and the longer leg is √3. Thus, tan(30°) = opposite/adjacent = 1/√3. Rationalizing the denominator gives us the correct value of √3/3.