Problem 7 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

\( \arctan \frac{\sqrt{3}}{3} = \pi /6 \)

Step 1: Understanding \( \arctan \)

The function \( \arctan \) is the inverse of the tangent function. It returns the angle whose tangent is the input. So, the task is essentially to find the angle whose tangent is \(\sqrt{3}/3\).

Step 2: Identifying Known Trigonometric Values

Common angles are in multiples of 30 degrees or \( \pi /6 \) radians and 45 degrees or \( \pi /4 \) radians. It is known that \( \tan(30) \) or \( \tan( \pi /6 \) equals to \(1/\sqrt{3}\) and \( \tan(45) \) or \( \tan( \pi /4 \) equals to 1.

Step 3: Evaluate \( \arctan(\sqrt{3}/3) \)

Since \( \tan(\pi/6) \) is \( 1/\sqrt{3} \), then \( \arctan(1/\sqrt{3}) \) will be \( \pi/6 \). Now consider that \( 1/\sqrt{3} \) is equal to \( \sqrt{3}/3 \). Therefore, \( \arctan(\sqrt{3}/3) \) equals to \( \pi /6 \).