Section 4.1

 A ray is a part of a line that has only one endpoint and extends forever in the opposite direction.  An angle is formed by two rays that have a common endpoint.  One ray is called the initial side and the other is the terminal side.  The endpoint of an angle’s initial and terminal side is the vertex of the angle.  An angle in standard position if 1. Its vertex is at the origin of a rectangular coordinate system 2. It’s initial side lies along the positive x-axis.

 Positive angles are generated by counterclockwise rotation.  Negative angles are generated by clockwise rotation.  An angle is called quadrantal if its terminal side lies on the x- or y-axis.  If a standard angle has a terminal side that lies in a quadrant, then we say the angle lies in that quadrant.

 One way to measure angles is in degrees. Count how much rotation a ray has completed of a 360˚ circle.  Names of angles:  Angles are often labeled using Greek letters such as:   Alpha  Beta  Theta,  = Omega,  = Phi

 Definition of a central angle – An angle whose vertex is at the center of the circle.  Definition of 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.

20 inches 5 inches

 Angles Formed by Revolution of Terminal Sides

90 ˚ -90 ˚ 90 ˚ 180 ˚ 270 ˚ 360 ˚ 0˚0˚0˚0˚ 0˚0˚0˚0˚ 0˚0˚0˚0˚ 180 ˚ -180 ˚ 270 ˚ -270 ˚ -360 ˚

Please do the following problems

 Two angles with the same initial sides and terminal sides but (possibly) different rotations are called coterminal angles.  An angle of x˚ (or x radians) is coterminal with angles of x˚ + k 360˚(or 2π) where k is an integer.

90 ˚ -90 ˚ 90 ˚ 180 ˚ 270 ˚ 360 ˚ 0˚0˚0˚0˚ 0˚0˚0˚0˚ 0˚0˚0˚0˚ 180 ˚ -180 ˚ 270 ˚ -270 ˚ -360 ˚

90 ˚ 180 ˚ 270 ˚ 360 ˚ 0˚0˚0˚0˚

See if you can figure out any short cuts while solving

 If a point is in motion on a circle of radius r, through and angle of  radians in time, t, then its linear speed is: where s is the arc length given by s = r θ, and its angular speed is:  The linear speed, v, of a point a distance r from the center of rotation is given by where ω is the angular speed in radians per unit of time.

(a) (b) (c) (d)

(a) (b) (c) (d)