Question

On a certain calculator, the inverse tangent function returns a value between $\mathbf{-}\mathbf{90}\mathbf{°}$ and$\mathbf{+}\mathbf{90}\mathbf{°}$. In what cases will this value correctly state the direction of a vector in the$\mathit{x}\mathit{y}$ plane, by giving its angle measured counter clockwise from the positive$\mathit{x}$ axis? In what cases will it be incorrect?

Short answer

This can be corrected by adding $180°$to the calculator's returned value.

Step-by-step solution

Definition of inverse tangent function.

The inverse tangent function, often known as arc tan, it is tangent function's inverse. It is used to figure out what angle a particular tangent makes.

Determine the vector direction.

A vector's direction is determined by the angle;

$\theta =ta{n}^{-1}\left(\frac{y-componentofvector}{x-componentofvector}\right)$

Analyse the graph. 

Defining the above graph.

The function will appropriately indicate the angle of a vector with respect to the positive x axis when the given vector is in the first and fourth quadrants, as shown in the diagram above.

When the provided vector is in the second or third quadrants, the function will not accurately express the angle of a vector with respect to the positive x axis.

Thus, this can be corrected by adding $180°$to the calculator's returned value.