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exact value of tan pi/6

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exact value of tan pi/6

Jun 26, 2017

#1
+7352
+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 !

Jun 27, 2017

#1
+7352
+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
+98130
+1

tan (pi /6)  =  1 / √3   =  √3 / 3

Jun 27, 2017