Find the exact value of tan (pi/3)

To find the value of tan(∏/3) let us consider a right-angled triangle. We know that tan (∏/3)= 60 degrees.

The tan function of an angle is equal to the length of the opposite side divided by the length of the adjacent side.

Tan θ = opposite side/ Adjacent Side

In terms of sine and cosine function, the tangent function is represented by

Tan θ = sin θ / Cos θ

To find the value of tan 60 degrees geometrically, consider an equilateral triangle ABC since each of an angle in an equilateral triangle is 600.

Therefore, ∠A = ∠B = ∠C = 60°

Draw a perpendicular line AD from A to BC.

Now consider the triangle, ABD and ADC,

We have, ∠ ADB = ∠ADC= 90° and

∠ ABD = ∠ACD= 60°

Therefore, AD=AD

According to AAS Congruency,

Δ ABD ≅ Δ ACD

From this, we can say

BD = DC

Let us take, AB = BC =2a

Then, BD= ½ (BC) =½ (2a) =a

By using Pythagoras theorem,

AB² = AD²– BD²

AD²= AB²-BD²

AD² =(2a)² – a²

AD² = 4a²-a²

AD² = 3a²

Therefore, AD=a√3

Now in triangle ADB,

Tan 600= AD/BD

= a√3/a = √3

Therefore, tan 60 degrees exact value is given by,

Tan 60=√3