Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.1 Part 2 Angles and Radian Measure Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Recognize and use the vocabulary of angles. Use degree measure. Use radian measure. Convert between degrees and radians. Draw angles in standard position. Find coterminal angles. Find the length of a circular arc. Use linear and angular speed to describe motion on a circular path.

Coterminal Angles Defintion: Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles. A complete rotation of a ray results in an angle measuring 360. Given angle θ, and continuing the rotation by a multiple of 360 will result in a different angle, θ + 360n,with the same terminal side: coterminal angles.

Coterminal Angles (continued) Calculating in Degrees Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360° results in a coterminal angle. Thus, an angle of θ° is coterminal with angles of θ° ± 360°n, where n is an integer. Calculating in Radians Increasing or decreasing the radian measure of an integer multiple of 2π results in a coterminal angle. Thus, an angle of θ radians is coterminal with angles of θ ± 2πn, where n is an integer.

Example: Finding Coterminal Angles (Degrees) Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following: a. 400° angle 400° – 360° = 40° b. –135° angle –135° + 360° = 225°

Your Turn Assume the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following: a. 525° angle b. –75° angle

Example: Finding Coterminal Angles (Radians) Assume the following angles are in standard position. Find a positive angle less than that is coterminal with each of the following: a. angle b. angle

Your Turn Assume the following angles are in standard position. Find a positive angle less than 2π that is coterminal with each of the following: a. angle b. angle

Arc Lengths Given a circle of radius “r”, any angle with vertex at the center of the circle is called a “central angle” The portion of the circle intercepted by the central angle is called an “arc” and has a specific length called “arc length” represented by “s” From geometry it is know that in a specific circle the length of an arc is proportional to the measure of its central angle For any two central angles θ1, and θ2, with corresponding arc lengths s1 and s2:

Arc Lengths Since this relationship is true for any two central angles and corresponding arc lengths in a circle of radius r: Let one angle be θ rad (θ1 = θ) with corresponding arc length s (s1 = s) and let the other central angle be 2π (θ2 = 2π) a whole rotation, with arc length 2πr (s2 = 2πr) the circumference of a circle.

Formula for Arc Length Let r be the radius of a circle and θ the nonnegative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is 𝑠=𝑟𝜃. **θ must be in radians**

Example: Finding the Length of a Circular Arc A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45°. Express arc length in terms of Then round your answer to two decimal places. We first convert 45° to radians:

Your Turn A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240.

Note Concerning Application Problems Involving Movement Along an Arc When a rope, chain, belt, etc. is attached to a circular object and is pulled by, or pulls, the object so as to rotate it around its center, then the length of the movement of the rope, chain, belt, etc. is the same as the length of the arc.

Example: Finding a Length A rope is being wound around a drum with radius .8725 ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72? Convert 39.72 to radian measure.

Your Turn Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225, through how many degrees will the larger gear rotate? Note: The motion of the small gear will generate an arc length on the small gear and an equal movement on the large gear

Definitions of Linear and Angular Speed

Linear Speed in Terms of Angular Speed

Example: Finding Linear Speed Long before iPods that hold thousands of songs and play them with superb audio quality, individual songs were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center. Before applying the formula we must express in terms of radians per second: Note: 1 revolution = 2π radians

Example: Finding Linear Speed (continued) A 45-rpm record has an angular speed of 45 revolutions per minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the record’s center. The angular speed of the record is radians per minute. The linear speed is

Your Turn A bicycle wheel with a radius of 12 inches is rotating at a constant rate of 3 revolutions every 4 seconds. a) What is the linear speed of a point on the rim of this wheel? b) What is the angular speed of a point on the rim of this wheel?

Your Turn A race car engine can turn at a maximum rate of 12 000 rpm. (revolutions per minute). What is the angular velocity in radians per second.