0.1: Angles and Radian Measure © 2008 Roy L. Gover ( Learning Goals: Define radians as another angle measure. Review angle measure Calculus Prerequisite Knowledge

Definition Initial Side Terminal Side Vertex Angle A is in standard position A x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves counter- clockwise, angle A is positive x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side moves clockwise, angle A is negative x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y

Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y Is this an angle?

Try This What kind of angle is this? (hint: where is the terminal side?) quadrantal angle

Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°

Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°

Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°

Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°

Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°

Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°

Try This What is the measure of this quadrantal angle? x y 0°

Try This If a 143° angle is in standard position, determine the quadrant in which the terminal side lies. x y 2

Try This If a 280° angle is in standard position, determine the quadrant in which the terminal side lies. x y 4

Important Idea There are two units of measure for angles: degrees: used in geometry radians: used in calculus In Precal, we use degrees and radians.

y x Radian: The length of the arc above the angle divided by the radius of the circle. Definition, in radians (rads)

y x Definition, in radians (rads) Unit Circle: the circle with radius of 1 unit If r =1, = s 1

Definition The radian measure of an angle is the distance traveled around the unit circle. Since circumference of a circle is 2 r and r =1, the distance around the unit circle is 2

Example Find the degree and radian measure of the angle in standard position formed by rotating the terminal side ½ of a circle in the positive direction. Leave your radian answer in terms of.

Example Find the degree and radian measure of the angle in standard position formed by rotating the terminal side 5/6 of a circle in the negative direction. Leave your radian answer in terms of.

Try This Find the degree and radian measure of the angle in standard position formed by rotating the terminal side 2/3 of a circle in the positive direction. Leave your radian answer in terms of.

Solution radians

Example  rads 360° rads 6.28

Example 45  (degrees) radians .785 radians

Example 90  (degrees) radians  1.57 radians

Try This  rads  180 rads  3.14

Try This  rads  -180 rads -- -3.14

Try This  rads  270 rads 4.71 Do you see a pattern?

Important Idea Radian measure allows the expansion of trig functions to model real-world phenomena where independent variables represent distance or time and not just an angle measure in degrees.

Important Idea If a circle contains 360° or 2  radians, how many radians are in 180° Use to change rads to degrees 180° rads Use to change degrees to rads rads 180°

Example Change 30° to radian measure in terms of .

Try This Change 120° to radian measure in terms of .

Try This Change 240° to radian measure in terms of .

Example Change radians to degree measure.

Example Change radians to degree measure. (hint: radians are not always stated in terms of .)

Try This Change radians to degree measure °

Try This Change -3.5 radians to degree measure to the nearest tenth °

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians

Definition The quadrantal angles in radians The terminal side is on an axis.

Definition Coterminal Angles: Angles that have the same terminal side. Important Idea In precal, angles can be larger than 360° or 2 radians.

Example Find positive angles and negative angles that are coterminal with 30°.

Important Idea To find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle.

Analysis 30° and 390° have the same terminal side, therefore, the angles are coterminal 30° x y x y 390°

Analysis 30° and 750° have the same terminal side, therefore, the angles are coterminal 30° x y x y 750°

Analysis 30° and 1110° have the same terminal side, therefore, the angles are coterminal 30° x y x y 1110°

Analysis 30° and -330° have the same terminal side, therefore, the angles are coterminal 30° x y x y -330°

How many angles can you find that are coterminal with a 30° angle and how do you find them? With Mrs. Love

Try This Find 3 angles coterminal with 60° 420°,780° and -300°

Try This Find one positive angle and two negative angle coterminal with radians. and,

Try This Find two positive angle and one negative angle coterminal with radians. and,

Lesson Close In your own words and without looking at your notes, write a definition for: Coterminal angle Radian

Practice 1.cws0-1/1-10 (slides 1-37) 2. cws/11-27 (slides 38-63)