In this section, weβll draw angles in standard position, convert between degrees and radians and degrees-minutes-seconds, find coterminal angles, and find the length of a circular arc.
Draw a line segment from the point \((5,-3)\) to the point \((-1,2)\text{.}\) Find the distance between those two points using the Pythagorean Theorem. Leave your answer in exact form.
A radian is a unit of measurement of angle size, and 1 radian is defined to be the angle that is made when the arc length for that angle on a circle of radius 1 is equal to 1. Check out this Desmos link and click the "play" button on the slider for a visual representation of this concept.
An angle is created by imagining two rays originating from the same point: the initial side of the angle is stationary and the terminal side is rotated around until the desired angle is created. Angles measured from initial to terminal side in a counterclockwise direction are positive. Angles measured from initial to terminal side clockwise are negative.
An angle is said to be in standard position if the initial and terminal sides of the angle meet at the origin and the initial side of the angle is along the positive x-axis.
Two angles are coterminal if their initial and terminal sides align, but they differ by an integer number of full rotations. In standard position, this means that the two angles are βin the same placeβ are different by some integer multiple of 360Β° (or \(2\pi\) radians).
An arc length is a distance along a part of the circumference of a circle. A formula for arc length is \(s = rΞΈ\text{,}\) where \(s\) stands for arc length along a circle of radius \(r\text{,}\) with an interior angle \(\theta\) centered at the center of the circle.
An angle drawn in standard position is indicated in two ways. Beginning with an initial side along the positive \(x\)-axis and ending with a terminal side in the second quadrant, the angle in the counter- clockwise direction is labeled \(theta\) and the angle in the clockwise direction is labeled \(alpha\text{.}\) The segment of the circumference of the circle between the initial and terminal sides of \(theta\) is the arc subtended by \(theta\text{,}\) and the length measure of that arc is the arc length for angle \(theta\text{.}\)
How big of an arc length will an angle of \(\dfrac{3\pi}{8}\) radians create on a circle of radius 6? Draw this angle and arc length on the circle of radius 6 provided.
