Chapter 4, Section 1

Chapter 4, Section 1
Radian Degree and Measure

• The word trigonometry is a word derived from Greek language meaning "measurement of triangles." It deals with the relationships among the sides and angles of triangles
  
  
• An angle is determined by rotating a ray about its endpoint
  

• The starting position of a ray is the initial side of the angle, and the position of the ray after rotation is the terminal side (shown below)
  
  


• Positive angles are generated by a counterclockwise rotation
• Negative angles are genereated by a clockwise rotation

Radian Measure

• One radian is the measure of a central angle (theta) that intercepts an arc (s) equal in length to the radius (r) of the circle


• Because the circumference of a circle is 2PIEr, it follows that a central angles of one full counterclockwise revolution corresponds to an arc length of s = 2PIEr. Therefore, 2PIE radians corresponds to 360, PIE radians corresponds to 180, and PIE over 2 radians corresponds to 90. Because 2PIE = about 6.28, there are just over six radius lengths in a full circle.
  
• In general, the radian measure of a central angle theta is obtained by dividing the arc length (s) by r. Therefore, s/r= theta, where theta is measured in radians.
  




• Because the radian measure of an angle of one full revolution is 2PIE, you can reason the following:

1/2 revolution = 2PIE/2 = PIE radians

1/4 revolution = 2PIE/4 = PIE/2 radians

1/6 revolution = 2PIE/6 = PIE/3 radians

• There are four quadrants in a coordinate system and are numbered I, II, III, and IV

Coterminal Angles

• Two angles are coterminal if they have the same initial and terminal sides

Example: The angles 0 and 2PIE are coterminal, as are the angles PIE/6 and 13PIE/6

• You can find that an angle is coterminal to a given angle theta by adding or subtracting 2PIE (one revolution)

Example: For the positive angle 13PIE/6, subtract 2PIE to obtain a coterminal angle

13PIE/6 - 2PIE = PIE/6

Degree Measure

• A second way to measure angles is in terms of degrees
  
• A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by PIE radians/180 degrees
   
2. To convert radians to degrees, multiply radians by 180 degrees/PIE radians

*PIE radians = 180 degrees

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Chapter 4, Section 2

Trigonometric Functions: The Unit Circle