Question

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

Answer 1

Answer on Question #61214 – Math – Trigonometry

Question

Evaluate $\tan 30{}^\circ$ without using a calculator by using ratios in a reference triangle.

Solution

Let ABC be an equilateral triangle with $AB = AC = BC = 1$.

Drop an altitude $AH$ from the top angle $A$, which cut the angle in half, dividing the equilateral triangle into two right triangles with a 30 degree angle. Because $AH$ is a median,

$$BH = HC = \frac{BC}{2} = \frac{1}{2}.$$

Using the Pythagorean Theorem

$$AH^2 + BH^2 = AB^2,$$ $$AH^2 = AB^2 - BH^2,$$ $$AH^2 = 1^2 - \left(\frac{1}{2}\right)^2 = 1 - \frac{1}{4} = \frac{3}{4},$$

find

$$AH = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}.$$

By definition, $\tan 30{}^\circ$ is given by

$$\tan 30{}^\circ = \tan \angle BAH = \frac{BH}{AH},$$ $$\tan 30{}^\circ = \frac{1}{2} : \frac{\sqrt{3}}{2} = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \approx 0.577.$$

Answer: $\frac{\sqrt{3}}{3}$.

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