Angles An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. A è B C Terminal Side Initial Side Vertex Unit 4 Trigonometric Functions 4.1 Angles and Their Measures (4.1)

Angles of the Rectangular Coordinate System 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. x y Terminal Side Initial Side Vertex   is positive Positive angles rotate counterclockwise. x y Terminal Side Initial Side Vertex   is negative Negative angles rotate clockwise.

Complete Student Checkpoint Draw each angle in standard position. a.30° b. 210° c. -120° d. 390° 30° 210° -120° 390°

Measuring Angles Using Degrees The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.  Acute angle 0º <  < 90º 90º Right angle 1/4 rotation  Obtuse angle 90º <  < 180º 180º Straight angle 1/2 rotation

Finding Complements and Supplements For an xº angle, the complement is a 90º – xº angle. Thus, the complement’s measure is found by subtracting the angle’s measure from 90º. For an xº angle, the supplement is a 180º – xº angle. Thus, the supplement’s measure is found by subtracting the angle’s measure from 180º.

In Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north. For example, the line of travel in this figure has the bearing of 155º

Degree-Minute-Second (DMS) measure 42º24'36" A degree, represented by the symbol º, is a unit of measure equal to 1/180 th of a straight angle. In the DMS system of angular measure, each degree is subdivided into: 60 minutes, denoted by, each minute is subdivided into: 60 seconds, denoted by,

Convert º to DMS. Convert 42º24'36" to degrees. Need to convert the decimal part to minutes and seconds. First convert to minutes: Convert the decimal part to seconds:

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. Radian Measure Consider an arc of length s on a circle or radius r. The measure of the central angle that intercepts the arc is  = s/r radians  O r s r

A central angle,, in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of ?

Conversion between Degrees and Radians  radians = 180º To convert degrees to radians: (degrees) To convert radians to degrees: (radians) Convert each angle in degrees to radians: 40º -160º 75º

Complete Student Checkpoint Convert each angle in degrees to radians. a. 60º b. 270º c Convert each angle in radians to degrees. a. b. c. 6 radians

Length of a Circular Arc Let r be the radius of a circle and  the non-negative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is: s = r   O s r A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2  /3 Solution:

Complete Student Checkpoint Find the perimeter of a 45º slice of a medium (6 in. radius) pizza. s = r 

Converting to Nautical Miles A nautical mile, represented by naut mi, is the length of 1 minute of arc along the Earth’s equator (radius ≈3956 statute miles). Applying the formula s = r  :

The distance from Boston to San Francisco is 2698 stat mi. How many nautical miles is it from Boston to San Francisco?

Angles and Their Measure