Question

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

Answer 1

Answer:

The value of $\tan 30\degree$ is $\frac{1}{\sqrt{3}}$

Step-by-step explanation:

In triangle ABC  (figure -1)

$\tan \theta =\frac{prependicular}{base}$

$\tan \theta =\frac{BC}{AB}$

$\tan 30\degree =\frac{1}{\sqrt{3}}$

Hence, the value of $\tan 30\degree$ is $\frac{1}{\sqrt{3}}$

Answer 2

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.