Draw an equilateral triangle with each equal side 's'.

Then Draw an altitude to any base .


Clearly,Tan60= p/b = altitude/(side of triangle/2)

[Becoz {'P'=altitude=(rt3/2)S }& {b=(side of tri/2)='s/2'}]


so,Tan60=(rt3/2)s/(s/2) [altitude=(rt3/2)×side]

=rt3

Understanding Tangent of 60 Degrees Geometrically
To find tan 60 degrees geometrically, we can use a special triangle known as the equilateral triangle.
Step 1: Construct an Equilateral Triangle
- Start with an equilateral triangle where each side measures 2 units.
- Label the vertices as A, B, and C.
Step 2: Draw the Altitude
- Draw a line segment from vertex A perpendicular to side BC. Let’s call the foot of the altitude point D.
- This altitude divides the equilateral triangle into two 30-60-90 right triangles.
Step 3: Analyze the 30-60-90 Triangle
- In a 30-60-90 triangle, the ratios of the lengths of the sides are well known:
- The side opposite the 30-degree angle (AD) is half the length of the hypotenuse (AB).
- The side opposite the 60-degree angle (BD) can be calculated using the properties of the triangle.
Step 4: Calculate the Side Lengths
- Since AB = 2, the side opposite the 30-degree angle (AD) is 1 (half of AB).
- Using the Pythagorean theorem, we find BD, which is the altitude:
- BD = sqrt(AB^2 - AD^2) = sqrt(2^2 - 1^2) = sqrt(4 - 1) = sqrt(3).
Step 5: Find Tan 60 Degrees
- Tan of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
- Tan 60 = BD/AD = sqrt(3)/1 = sqrt(3).
Thus, we conclude that tan 60 degrees equals sqrt(3), derived geometrically from the properties of a 30-60-90 triangle.