Trigonometric Functions Angles and Radian Measures Objectives Identifying
Trigonometric Functions Angles and Radian Measures
Objectives § § § § Identifying the parts of an angle Measuring angles using degrees and radians Calculating radian measure Conversion between degrees and radians Drawing angles in standard position Finding coterminal angles Calculating the length of a circular arc
Vocabulary Acute angle Angle Coterminal angles Initial side Negative angle Obtuse angle Positive angle Quadrantal angle Radian Measure Ray Reflex angle Right angle Quadrant Terminal side Vertex
Angles A ray is a part of a line that has only one endpoint and extends forever in one direction. Te rm in al si de An angle is formed by two rays that have a common endpoint. The common endpoint is called the vertex. One ray is called the initial side, and the other ray is called the terminal side. Initial side
An arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. Several methods can be used to name an angle. B 1 A C Ø By using the end points with the vertex in the middle L CAB L BAC Ø By using the letter of the vertex or a number LA L 1
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. min ide al s When an angle is in standard position, its terminal side can lie in a quadrant. If the terminal side lies on the x-axis or y-axis, it does not lie in any quadrant. In this case it is called a quadrantal angle. Ter There are two types of rotation: (1) a counterclockwise rotation which results in a positive angle and (2) a clockwise rotation which yields a negative angle. Positive x-axis Initial side
Measuring Angles Using Degrees We use the symbol º for degrees to indicate the measure of an angle. Certain angles have certain names based on their measures.
Relationship Between Degrees and Radians One complete rotation measures 360º.
Conversion Between Degrees and Radians
Positive angles Negative angles Degree and Angle Measures of Selected Positive and Negative Angles
Drawing Angles in Standard Position II I Which quadrant does each angle lie in? Quadrant III IV Quadrant IV
Coterminal Angles
Initial side al s min Ter
ide ls na rm i Te Initial side
Find a positive angle that is less than 360º that is coterminal with each of the following: 420º angle − 120º angle 400º angle − 135º angle
The Length of a Circular Arc
Linear Speed ▪ The second hand of a clock is 8 centimeters long. Find the linear speed of the tip of this second hand.
Angular Speed The circular blade on a saw rotates at 2400 revolutions per minute. Find the angular speed in radians per second.
- Slides: 28