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Chapter 4: Problem 13

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

Short Answer

Expert verified
-pi/3

Step by step solution

01

Understanding the arctan function

The arctan function, also known as the inverse tangent function, is the inverse of the tangent function. It is used to obtain an angle from the tangent of that angle. The angle it gives will always lie in the interval \(-\pi/2\) to \(\pi/2\) (or \(90掳\) to \(-90掳\)). Also, the tangent of an angle in the unit circle is positive in the first and third quadrants, and negative in the second and fourth quadrants.
02

Identify the given value in the unit circle

We are given that the value of the tangent is \(-\sqrt{3}\). The square root of 3 is associated with an angle of \(\pi/3\) (or 60掳) in the unit circle. Since the value is negative, this corresponds to an angle in the second or fourth quadrant.
03

Determine the correct quadrant

Since arctan gives values from \(-\pi/2\) to \(\pi/2\), this corresponds to the fourth quadrant. Therefore, the angle we are seeking is the negative version of \(\pi/3\), which is \(-\pi/3\).

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Most popular questions from this chapter

Sketch a graph of the function. $$h(v)=\arccos \frac{v}{2}$$

Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse trigonometric function.

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