24 FINANCE, SAVING, AND INVESTMENT © 2012 Pearson Addison-Wesley.
Slide 5.3- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
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Transcript of Slide 5.3- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
OBJECTIVES
Trigonometric Functions of Any Angle
Learn and use the definitions of the trigonometric functions of any angle.Learn and use the signs of the trigonometric functions.Learn to find and use a reference angle.Learn to find the area of an SAS triangle.Learn and use the unit circle definitions of the trigonometric functions.Learn and use some basic trigonometric identities.
SECTION 5.3
1
2
3
4
5
6
Slide 5.3- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
Slide 5.3- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
sin y
r
cos x
r
tan y
x, x 0
csc r
y, y 0
sec r
x, x 0
cot x
y, y 0
Let P(x, y) be any point on the terminal ray of an angle in standard position (other than the
r x2 y2 .origin), and let Then r > 0, and:
Slide 5.3- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1 Finding Trigonometric Function Values
Suppose that is an angle whose terminal side contains the point P(–1, 3). Find the exact values of the six trigonometric functions of .
Solution
r2 x2 y2
1 2 32
10
Slide 5.3- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1 Finding Trigonometric Function Values
Solution continued
sin y
r
3
10
3 10
10
cos x
r
1
10
10
10
tan y
x
3
1 3
csc r
y
10
3
10
3
sec r
x
10
1 10
cot x
y
1
3
1
3
Now, with x 1, y 3 and r 10 we have
Slide 5.3- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES
These equations hold for any integer n.
in degrees
sin sin n360º
cos cos n360º
in radians
sin sin 2n
cos cos 2n
Slide 5.3- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
TRIGONOMETRIC FUNCTION VALUES OF QUADRANTAL ANGLES
00º 0 1 0 und. 1 und.
deg
sin cos tan csc sec tanradians
180º 0 1 0 und. 1 und.
2360º 0 1 0 und. 1 und.
32
270º 1 0 und. 0 und. 1
2
90º 1 0 und. 0 und. 1
Slide 5.3- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SIGNS OF TRIGONOMETRIC FUNCTIONS
Slide 5.3- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF A REFERENCE ANGLE
Let be an angle in standard position that is not a quadrantal angle. The reference angle for is the positive acute angle ´(“theta prime”) formed by the terminal side of and the positive or negative x-axis.
Slide 5.3- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF A REFERENCE ANGLE
Slide 5.3- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
DEFINITION OF A REFERENCE ANGLE
Slide 5.3- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Evaluating Trigonometric Functions
Given that tan 3
2, and cos 0, find the
exact value of sin and sec.
Solution
tan º 0 and cos 0, lies in Quandrant III
x and y are both negative
tan y
x
3
2
3
2
r x2 y2 2 2 3 2 4 9 13
Slide 5.3- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4 Evaluating Trigonometric Functions
Solution continued
With x 2, y 3, and r 13 we can find
sin and sec.
sin y
r
3
13
3 13
13
sec r
x
13
2
13
2
Slide 5.3- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC
FUNCTION VALUESStep 1 If the degree measure of is greater
than 360º, then find a coterminal angle for with degree measure between 0º and 360º. Otherwise, use in Step 2.
Step 2 Find the reference angle ´ for the angle resulting in Step 1. Write the trigonometric function of the acute angle, ´.
Slide 5.3- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC
FUNCTION VALUESStep 3 The sign of a trigonometric function of
depends on the quadrant in which lies. Use the signs of the trigonometric functions to determine when to change the sign of the associated value for ´. (Since ´ is an acute angle, all its function values are positive.)
Slide 5.3- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6Using the Reference Angle to Find Values of the Trigonometric Function
a. tan 330º
360º 330º 30º
tan tan 30º 3
3
b. sec59
6
Find the exact value of each expression.
Solution
Step 1 0º < 330º < 360º, find its reference angle
Step 2 330º is in Q IV, its reference angle ´ is
Slide 5.3- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6Using the Reference Angle to Find Values of the Trigonometric Function
tan 330º tan 30º3
3
Solution continued
Step 3 In Q IV, tan is negative, so
596
11 48
6
116
8b. Step 1
116
is between 0 and 2π coterminal with59
6
Slide 5.3- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 6Using the Reference Angle to Find Values of the Trigonometric Function
sec59
6sec
116
sec6
2 3
3
116
Solution continued
Step 3 In Q IV, sec > 0, so
Step 2
2 11
6
6
sec sec6
2 3
3
is in Q IV, its reference angle ´ is
Slide 5.3- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
AREA OF A TRIANGLE
In any triangle, if is the included angle between sides b and c, the area K of the triangle is given by
K 1
2bcsin
Slide 5.3- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8 Finding a Triangular Area Determined by Cellular Telephone Towers
Three cell towers are set up on three mountain peaks. Suppose the lines of sight from tower A to towers B and C form an angle of 120º, and the distances between tower A and towers B and C are 3.6 miles and 4.2 miles, respectively. Find the area of the triangle having these three towers as vertices.
Slide 5.3- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7 Measuring the Height of Mount Kilimanjaro
SolutionArea of the triangle with angle = 120º included between sides of lengths b = 3.6 and c = 4.2 is
A 1
2bcsin
1
23.6 4.2 sin120º
1
23.6 4.2 sin 60º
1
23.6 4.2 3
26.55 square miles
Finding a Triangular Area Determined by Cellular Telephone Towers
Slide 5.3- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
Slide 5.3- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
Let t be any real number and let P(x, y) be the point on the unit circle associated with t. Then
sin t y
cos t x
tan t y
x, x 0
csc t 1
y, y 0
cot t x
y, y 0
sec t 1
x, x 0
Slide 5.3- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
BASIC TRIGONOMETRIC IDENTITIES
Quotient Identities
tan t sin t
cos t cos2 t sin2 t 1
csc t 1
sin t
cot t cos t
sin t
sec t 1
cos t
1 tan2 t sec2 t
1 cot2 t csc2 t
Reciprocal Identities
Pythagorean Identities
Slide 5.3- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities
a. Given sin t 1
3 and cos t 0, find cos t and tan t.
b. Given sec t 2 and tan t 0, find tan t.
cos2 t sin2 t 1
cos2 t 1
3
2
1
a. Use Pythagorean identity involving sin t.
Solution
cos2 t 1 1
9
Slide 5.3- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities
Solutioncos2 t
8
9
cos t 8
9
2 2
3
cos t 2 2
3cos t 0 is given
tan t sin t
cos t
13
2 2
3
1
2 2
2
4
Slide 5.3- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 9Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities
1 tan2 t sec2 t
1 tan2 t 2 2
tan2 t 3
tan t 3
tan t 3 tan t 0 is given
Use Pythagorean identity involving sec t.
Solution continuedb. Given sec t 2 and tan t 0, find tan t.