Chapter 4 Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 4.2 Trigonometric Functions: The Unit Circle
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Chapter 4Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
4.2 Trigonometric Functions: The Unit Circle
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
Objectives:
• Use a unit circle to define trigonometric functions of real numbers.
• Recognize the domain and range of sine and cosine functions.
• Find exact values of the trigonometric functions at• Use even and odd trigonometric functions.• Recognize and use fundamental identities.• Use periodic properties.• Evaluate trigonometric functions with a calculator.
.4
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3
The Unit Circle
A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is 2 2 2.x y r
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4
The Unit Circle (continued)
In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5
The Six Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6
Definitions of the Trigonometric Functions in Terms of a Unit Circle
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7
Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.
sin t y
cos t x
tany
tx
12
32
112
3 32
1 3 333 3
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8
Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.
1csct
y 1
sectx
cotx
ty
12
12
1 2
3 32
2 3 2 333 3
32 312
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9
The Domain and Range of the Sine and Cosine Functions
y = cos x y = sin x
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10
Exact Values of the Trigonometric Functions at
Trigonometric functions at occur frequently. We use
the unit circle to find values of the trigonometric functions at
We see that point P = (a, b)
lies on the line y = x. Thus, point P
has equal x- and y-coordinates: a = b.
4t
4t
.4
t
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11
Exact Values of the Trigonometric Functions at (continued)
We find these coordinates as follows:
4t
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12
Exact Values of the Trigonometric Functions at (continued)
We have used the unit circle to find the coordinates of point
P = (a, b) that correspond to
4t
.4
t
2 2,
2 2P
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13
Example: Finding Values of the Trigonometric Functions
4t
at
2 2,
2 2P
Find csc4
1csc
4 y 1
22
2
2
2 22
2
Find sec4
1sec
4 x
1
22
2
2
2 22
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14
Example: Finding Values of the Trigonometric Functions
4t
at
2 2,
2 2P
Find cot4
cot4
xy
222
2
1
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15
Trigonometric Functions at 4
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16
Even and Odd Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17
Example: Using Even and Odd Functions to Find Values of Trigonometric Functions
Find the value of each trigonometric function:
sec4
sin4
sec4
2
sin4
2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18
Fundamental Identities
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19
Example: Using Quotient and Reciprocal Identities
Given and find the value of each of
the four remaining trigonometric functions.
2sin
3t 5
cos3
t
sintan
cost
tt
235
3
1csc
sint
t 1 3
2 23
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20
Example: Using Quotient and Reciprocal Identities (continued)
Given and find the value of each of
the four remaining trigonometric functions.
2sin
3t 5
cos3
t
1sec
cost
t
1 3
5 53
1cot
tant
t 1 5
2 5 2 55
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21
The Pythagorean Identities
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22
Example: Using a Pythagorean Identity
Given that and find the value of
using a trigonometric identity.
0 ,2
t 1
sin2
t
cos t
2 2sin cos 1t t
221
cos 12
t
21cos 1
4t
2 1cos 1
4t
2 3cos
4t
3 3cos
4 2t
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23
Definition of a Periodic Function
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24
Periodic Properties of the Sine and Cosine Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25
Periodic Properties of the Tangent and Cotangent Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26
Example: Using Periodic Properties
Find the value of each trigonometric function:
5cot
4
9cos
4
cot4
cot
4 1
9cos
4
cos 2
4
cos
4 2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27
Repetitive Behavior of the Sine, Cosine, and Tangent Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28
Using a Calculator to Evaluate Trigonometric Functions
To evaluate trigonometric functions, we will use the keys on a calculator that are marked SIN, COS, and TAN. Be sure to set the mode to degrees or radians, depending on the function that you are evaluating. You may consult the manual for your calculator for specific directions for evaluating trigonometric functions.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29
Example: Evaluating Trigonometric Functions with a Calculator
Use a calculator to find the value to four decimal places:
sin4
csc1.5
0.7071
1.0025
Related Documents
