Demana Waits Foley Kennedy 4 3 Trigonometric Functions

Demana, Waits, Foley, Kennedy 4. 3 Trigonometric Functions of Any Angle Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1

What you’ll learn about n n Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16 -point unit circle … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2

Initial Side, Terminal Side Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 3

Positive Angle, Negative Angle Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 4

Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 5

Example: Finding Coterminal Angles Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 6

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 7

Example: Finding Coterminal Angles Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 8

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9

Example: Evaluating Trig Functions Determined by a Point in QI Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 10

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 11

Trigonometric Functions of any Angle Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 12

Reference Angles Reference Angle For any given angle, its reference angle is an acute version of that angle. In standard position, the reference angle is the smallest angle between the terminal side and the x-axis. The values of the trig functions of angle θ are the same as the trig values of the reference angle for θ, give or take a minus sign. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 13

Evaluating Trig Functions of a Nonquadrantal Angle θ 1. 2. 3. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 14

Evaluating Trig Functions of a Nonquadrantal Angle θ 4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. 5. Use the coordinates of point P and the definitions to determine the six trig functions. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 15

Example: Evaluating More Trig Functions Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 16

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 17

Example: Using One Trig Ratio to Find the Others Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 18

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 19

Solution Continued Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 20

Unit Circle The unit circle is a circle of radius 1 centered at the origin. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 21

Trigonometric Functions of Real Numbers Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 22

Periodic Function Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 23

The 16 -Point Unit Circle Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 24