Unit 5 Angles and Radian Measure 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. Radian Measure as a Distance on the Unit Circle

A Sense of Angle Sizes See if you can guess the size of these angles in degrees You will be working so much with these angles

Angles An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.

An angle is in standard position if Angles (continued) An angle is in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side lies along the positive x-axis.

Angles (continued) When we see an initial side and a terminal side in place, there are two kinds of rotations that could have generated the angle. Positive angles are generated by counterclockwise rotation. Thus, angle is positive. Negative angles are generated by clockwise rotation. Thus, angle is negative.

Angles (continued) An angle is called a quadrantal angle if its terminal side lies on the x-axis or on the y-axis. Angle is an example of a quadrantal angle.

Two angles with the same initial and terminal sides but Coterminal Angles Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles.

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

Measuring Angles Using Degrees Angles are measured by determining the amount of rotation from the initial side to the terminal side. A complete rotation of the circle is 360 degrees, or 360°. An acute angle measures less than 90°. A right angle measures 90°. An obtuse angle measures more than 90° but less than 180°. A straight angle measures 180°. Think about the direction and amount of rotation an angle has is a circle.

Lets Estimate degrees as a fraction of a circle

Suppose a ray is rotated about its endpoint Suppose a ray is rotated about its endpoint. Give the degree measure of the angle formed if the ray makes:   360° 300° 900°

The degree measure of an angle is given The degree measure of an angle is given. Determine its fractional part of a complete rotation. 180o 90o 45o 270o        

Complete Figuring out all the Angles Task Part A

Day 2 Warm-up Marissa is running a race on a circular track that has a radius of 200 feet. She is competing against 2 other runners. The coach is at the center and notes each racer’s location after 30 seconds.  

Is the Degree the only unit of measurement for angles in a circle? NO! What else? Radians What is a Radian? One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle. Huh?? What?? Why??

Some definitions first….. A central angle of a circle is an angle with a vertex at the center of the circle. An intercepted arc is the arc that is “captured” by the central angle.

Radians r When the central angle intercepts an arc that has the same length as a radius of the circle, the measure of the angle is defined as a radian. r Like degrees, radians measure the amount of rotation from the initial side to the terminal side of the angle.

Break down…… Where does π come from? How many radians (like radius) in a full circle?? How many radians in a half circle?? Number of times the length of the diameter fits around the a circle. Hence C=D π 2π….or 6.28 and a little more π….or 3.14 and a little more

So what your saying is…… A full circle is 360 ° Or we could say a full circle is 2π Radians. A half of a circle is 180 ° Or we could say a half circle is π Radians.

A Sense of Angles Revisited See if you can guess the size of these angles first in degrees and then in radians. You will be working so much with these angles, you should know them in both degrees and radians.

Complete Figuring out all Angles Task Part B

Closure: Convert each angle in degrees to radians: a. 60°= b. 270°=      

Example: Converting from Radians to Degrees Convert each angle in radians to degrees: a) b) c)          

Degree and Radian Measures of Angles Commonly Seen in Trigonometry In the figure below, each angle is in standard position, so that the initial side lies along the positive x-axis.