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Chapter 4: Problem 9

Evaluate the expression without using a calculator. $$ \arctan \frac{\sqrt{3}}{3} $$

Short Answer

Expert verified
The expression \( \arctan \frac{\sqrt{3}}{3} \) evaluates to 30 degrees or \( \frac{\pi}{6} \) radians.

Step by step solution

01

Identify the trigonometric function

The exercise provides an arctan function. This represents the inverse tangent function. It is used to calculate the angle whose tangent is a given value. In this case, the given value is the fraction \( \frac{\sqrt{3}}{3} \). The goal is to find an angle whose tangent equals this fraction.
02

Recognize the standard triangles

This fraction, \( \frac{\sqrt{3}}{3} \), is equivalent to \( \frac{1}{\sqrt{3}} \). This is the tan of 30 degrees, or \( \frac{\pi}{6} \) radians in a standard 30-60-90 degrees triangle.
03

Evaluate the expression

Therefore, the expression \( \arctan \frac{\sqrt{3}}{3} \) evaluates to 30 degrees, or \( \frac{\pi}{6} \) radians.

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