Question

Evaluate tan 30° without using a calculator by using ratios in a reference triangle. Please Explain!

Answer 1

tan 30 degrees would be equivalent to sin(30)/cos(30).
Sin(30) is 1/2, and cos(30) is (3)^(1/2)/2.
((1/2)/((3^1/2)/2) can be made easier to solve by taking the reciprocal of the denominator, and inverting it.
This leaves (1/2)*(2/(3^(1/2)).
The 2 in the denominator of sin(30) cancels the 2 in the numerator of cos(30).
Leaving (1/1)*(1/(3^1/2)).
Here, you can see that tan(30) equals 1/(3^(1/2)).

In regards to triangles, sin(30) is referring to a 30 degree angle in a 1-(3)^(1/2)-2 triangle, where sin is opposite/hypotenuse, and, so, is opposite the smaller, 1, leg.
As such, cos(30) is adjacent/hypotenuse and would be adjacent to the 3^(1/2), larger, leg. The hypotenuse is the same in both instances, in order to accommodate the Pythagorean theorem.

Answer 2

The value of
`\tan 30\degree` is
`(1)/(√(3))`

Explanation:

In triangle ABC (figure -1)

`\tan \theta =(prependicular)/(base)`

`\tan \theta =(BC)/(AB)`

`\tan 30\degree =(1)/(√(3))`

Hence, the value of
`\tan 30\degree` is
`(1)/(√(3))`