Evaluate the 𝑎𝑟𝑐𝑡𝑎𝑛(−⋅ √3). A) -30o B) -60o C) 30o D) 60o

Answer 1

-60 degrees = -pi/3 radians

arctan(-sqr3) = the angle whose tangent is -sqr3

tan(-60) = - sqr3

Answer 2

Sure, let's break it down:

1. The arctangent function, often denoted as atan or tan^(-1), is the inverse of the tangent function. It returns the angle whose tangent is a given number.

2. In this case, we're looking for the angle whose tangent is -√3. 

3. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. 

4. In the unit circle, an angle of 60° or π/3 radians has a tangent of √3, and an angle of -60° or -π/3 radians has a tangent of -√3. 

5. Therefore, the arctangent of -√3 is -60° or -π/3 radians.

6. This is because the arctangent function returns the angle in the interval from -π/2 to π/2 radians, or -90° to 90°. 

7. So, the correct answer is B) -60°.

Alternatively, the tangent function is defined as the ratio of the sine function to the cosine function for a given angle. In mathematical terms, if we have an angle 'a', we can write:

tan(a) = sin(a) / cos(a)

Now, if we consider an angle of -60 degrees:

tan(-60°) = sin(-60°) / cos(-60°)

The sine of -60 degrees is -√3/2 and the cosine of -60 degrees is 1/2. If we substitute these values into the equation, we get:

tan(-60°) = (-√3/2 )/ (1/2)

Simplifying this, we find:

tan(-60°) = -√3

So, the tangent of -60 degrees is -√3, which is the ratio of the sine of -60 degrees to the cosine of -60 degrees.

This last solution requires you to know intimately how the unit circle works and the formulas of the typical angles in trigonometry.