+0

# exact value of tan pi/6

0
87
2

exact value of tan pi/6

Guest Jun 26, 2017

#1
+4174
+2

First, let's draw an equilateral triangle where the sides are length  n  and the angles are  pi/3  .

Second, draw a line that bisects the angle. Since the other two sides are the same length, it also bisects the opposite side and forms a right angle. This forms the angles   (pi/3)/2  =  pi/6

Now, let's look at this third triangle. Let's call the length of the remaining side  " a " .

We can find  " a "  using the Pythagorean theorem.

a2 + (n/2)2  =  n2

a  = $$\sqrt{n^2-(\frac{n}{2})^2}=\sqrt{\frac{4n^2-n^2}{4}}=\frac{\sqrt3n}{2}$$

tan( pi/6 )  =  opposite / adjacent  =  $$\frac{n}{2}\,/\,\frac{\sqrt3n}{2}=\frac{n}{2}\,*\,\frac2{\sqrt3n}=\frac1{\sqrt3}=\frac{\sqrt3}{3}$$

No calculator needed !

hectictar  Jun 27, 2017
Sort:

#1
+4174
+2

First, let's draw an equilateral triangle where the sides are length  n  and the angles are  pi/3  .

Second, draw a line that bisects the angle. Since the other two sides are the same length, it also bisects the opposite side and forms a right angle. This forms the angles   (pi/3)/2  =  pi/6

Now, let's look at this third triangle. Let's call the length of the remaining side  " a " .

We can find  " a "  using the Pythagorean theorem.

a2 + (n/2)2  =  n2

a  = $$\sqrt{n^2-(\frac{n}{2})^2}=\sqrt{\frac{4n^2-n^2}{4}}=\frac{\sqrt3n}{2}$$

tan( pi/6 )  =  opposite / adjacent  =  $$\frac{n}{2}\,/\,\frac{\sqrt3n}{2}=\frac{n}{2}\,*\,\frac2{\sqrt3n}=\frac1{\sqrt3}=\frac{\sqrt3}{3}$$

No calculator needed !

hectictar  Jun 27, 2017
#2
+75352
+1

tan (pi /6)  =  1 / √3   =  √3 / 3

CPhill  Jun 27, 2017

### 9 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details