Question

give the degree measure of \\( \theta \\) if it exists. do not use a calculator.

\\( \theta=\arctan \left(\frac{\sqrt{3}}{3}\
ight) \\)

select the correct choice below and fill in any answer boxes in your choice

\\( \bigcirc \\) a. \\( \theta=\square^{\circ} \\)
(type your answer in degrees.)

\\( \bigcirc \\) b.
\\( \arctan \left(\frac{\sqrt{3}}{3}\
ight) \\) does not exist.

Explanation:

Step1: Recall the value of tangent

We know that \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\).

Step2: Use the property of arctangent

Since \(y = \arctan(x)\) is the inverse function of \(y=\tan(x)\) for \(x\in(-\frac{\pi}{2},\frac{\pi}{2})\) (in radians) or \(x\in(- 90^{\circ},90^{\circ})\) (in degrees), and \(\tan\theta=\frac{\sqrt{3}}{3}\) when \(\theta = 30^{\circ}\) (because \(30^{\circ}\in(-90^{\circ},90^{\circ})\)).

Answer:

A. \(\theta = 30^{\circ}\)