Angles and their Measures

1. Angles and their Measures Convert between degrees minutes seconds and decimal forms of angles Find the arc length Convert from degrees to radians Find the area of a sector of a circle Find the linear speed of an object traveling in circular motion

2. Angles A ray is that portion of a line that starts at a point on the line and extends indefinitely in on direction The starting point of the line is called the vertex If two rays are drawn with a common vertex they form an angle We call one of the rays of an angle the initial side and the other the terminal side

3. Continued • The angle that is formed is identified by showing the direction and amount of rotation from the initial side to the terminal side. • A rotation that counterclockwise is positive • A rotation that is clockwise is negative We use lower case Greek letters to identify these angles

4. An angle θ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x axis.

5. When an angle is in standard position, the terminal side will lie either in quadrant in which case we say that the angle lies in that quadrant If the angle lies on the x axis or y axis then we say that the angle is a quadrantal angle

6. Smaller than one degree Subdivisions of a degree can be expressed by using a decimal we also will use notation minutes and seconds One minute is defined as of a degree and one second is

7. Converting between Degrees minutes seconds and decimals Convert 50°6’21” Convert 21.256 to the DMS form

8. Central Angles A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle intersects an arc on the circle

9. Radians If the radius of the circle is r and the length of the arc subtended by the central angle is also r, then the measure of the angle is 1 radian. For a circle of radius 1 the rays of a central angle with measure 1 radian would intersect an arc length 1

10. Relationship between Degrees and Radians 1 revolution = 2Π 180°= Π radians 1 degree= radians 1 radian = degrees

11. Examples Convert 60° to radians Convert to degrees

12. Commonly encountered angles

13. Arc length For a circle of radius r a central angle of θ radians subtends an arc whose length s is s=rθ ONLY RADIANS Degrees

14. Area of a sector The area A of a sector of a circle of radius r formed by a central angle θ radians is A = r2θ

15. Circular Motion Linear Speed Last quarter: The average speed of an object as the distance traveled divided by the elapsed time Suppose an object travels around a circle of radius r at a constant speed. If s is the distance traveled in time t around this circle then the LINEAR SPEED

16. Circular Motion Angular Speed As this object travels around the circle suppose that θ (only measured in radians) is the central angle swept out of time t. The angular speed ω (omega) of this object is the angel swept out divided by the elapsed time