Key Concepts & Glossary

Key Equations

arc length	[latex]s=r\theta[/latex]
area of a sector	[latex]A=\frac{1}{2}\theta {r}^{2}[/latex]
angular speed	[latex]\omega =\frac{\theta }{t}[/latex]
linear speed	[latex]v=\frac{s}{t}[/latex]
linear speed related to angular speed	[latex]v=r\omega[/latex]

An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle.
An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise.
To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle represents.
In addition to degrees, the measure of an angle can be described in radians.
To convert between degrees and radians, use the proportion [latex]\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }[/latex].
Two angles that have the same terminal side are called coterminal angles.
We can find coterminal angles by adding or subtracting 360° or [latex]2\pi[/latex].
Coterminal angles can be found using radians just as they are for degrees.
The length of a circular arc is a fraction of the circumference of the entire circle.
The area of sector is a fraction of the area of the entire circle.
An object moving in a circular path has both linear and angular speed.
The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time.
The linear speed of an object traveling along a circular path is the distance it travels in a unit of time.

angular speed: the angle through which a rotating object travels in a unit of time

area of a sector: area of a portion of a circle bordered by two radii and the intercepted arc; the fraction [latex]\frac{\theta }{2\pi }[/latex] multiplied by the area of the entire circle

coterminal angles: description of positive and negative angles in standard position sharing the same terminal side

degree: a unit of measure describing the size of an angle as one-360th of a full revolution of a circle

linear speed: the distance along a straight path a rotating object travels in a unit of time; determined by the arc length

measure of an angle: the amount of rotation from the initial side to the terminal side

negative angle: description of an angle measured clockwise from the positive x-axis

positive angle: description of an angle measured counterclockwise from the positive x-axis

radian measure: the ratio of the arc length formed by an angle divided by the radius of the circle

radian: the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle

ray: one point on a line and all points extending in one direction from that point; one side of an angle

reference angle: the measure of the acute angle formed by the terminal side of the angle and the horizontal axis

standard position: the position of an angle having the vertex at the origin and the initial side along the positive x-axis