How do you find the exact value of #tan(pi/6)#?

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Answer 1 · 2016-04-08T10:22:31

#tan(pi/6)=1/sqrt(3)=sqrt(3)/3#

(see image below)
enter image source here
This is one of the standard trigonometric triangles.

#sqrt(3)# has been determined using Pythagorean Theorem.

Sum of interior angles of a triangle is always #pi# radians.

Answer 2 · 2016-04-08T10:24:08

#tan(pi/6) = sqrt3/3 approx 0.577#

Using the identity

#tan = sin/cos#,

and

#sin(pi/6) = sin(30) = 1/2#
#cos(pi/6) = cos(30) = sqrt3/2#

then

#tan(pi/6) = (1/2)/(sqrt3/2)#.

You should know that dividing by one number is the same as multiplying by its reciprocal, so

#(1/2)/(sqrt3/2) = 1/2 * 2/sqrt3#

Cancelling the #2#'s and rationalising the denominator,

#1/2 * 2/sqrt3 = 1/sqrt3#
#1/sqrt3 * sqrt3/sqrt3 = sqrt3/3#

Therefore,

#tan(pi/6) = sqrt3/3#

Using a calculator,

#tan(pi/6) approx 0.577#