AP®︎ Precalculus: Topic 3.13 Practice Test

Essential Knowledge: 3.13.A.1

The polar coordinate system is based on a grid of circles centered at the origin and on lines through the origin. The positive $x$-axis is called the polar axis. Polar coordinates of a point are defined as an ordered pair, $(r, \theta)$. The value of $\theta$ is given by the measure of an angle in standard position whose terminal ray lies in a line that passes through the point, and $r$ is the radial displacement of the point from the origin. Positive values of $r$ indicate radial displacement from the origin along the terminal ray of the angle $\theta$. Negative values of $r$ indicate radial displacement from the origin along the line containing the terminal ray of the angle $\theta$—but in the opposite direction of the terminal ray of the angle $\theta$. In the polar coordinate system, the same point can be represented many ways with combinations of positive and negative values of $r$ and $\theta$.

Essential Knowledge: 3.13.A.2

The coordinates of a point in the polar coordinate system, $(r, \theta)$, can be converted to coordinates in the rectangular coordinate system, $(x, y)$, using $x = r \cos \theta$ and $y = r \sin \theta$.

Essential Knowledge: 3.13.A.3

The coordinates of a point in the rectangular coordinate system, $(x, y)$, can be converted to coordinates in the polar coordinate system, $(r, \theta)$, using $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\left(\frac{y}{x}\right)$ for $x > 0$ or $\theta = \arctan\left(\frac{y}{x}\right) + \pi$ for $x < 0$.

Essential Knowledge: 3.13.A.4

A complex number can be understood as a point in the complex plane and can be determined by its corresponding rectangular or polar coordinates. When the complex number has the rectangular coordinates $(a, b)$, it can be expressed as $a + bi$. When the complex number has polar coordinates $(r, \theta)$, it can be expressed as $(r \cos \theta) + i(r \sin \theta)$.