Transcript 6.1 The Trigonometric Functions of Acute Angles

CHAPTER 6:
The Trigonometric Functions
6.1
6.2
6.3
6.4
6.5
6.6
The Trigonometric Functions of Acute Angles
Applications of Right Triangles
Trigonometric Functions of Any Angle
Radians, Arc Length, and Angular Speed
Circular functions: Graphs and Properties
Graphs of Transformed Sine and Cosine
Functions
Copyright © 2009 Pearson Education, Inc.
6.1
Trigonometric Functions of Acute Angles




Determine the six trigonometric ratios for a given
acute angle of a right triangle.
Determine the trigonometric function values of 30º,
45º, and 60º.
Using a calculator, find function values for any acute
angle, and given a function value of an acute angle,
find the angle.
Given the function values of an acute angle, find the
function values of its complement.
Copyright © 2009 Pearson Education, Inc.
Right Triangles and Acute Angles
An acute angle is an angle with measure greater than
0º and less than 90º.
Greek letters such as  (alpha),  (beta),  (gamma), 
(theta), and  (phi) are often used to denote an angle.
We label the sides with respect to angles. The
hypotenuse is opposite the right angle. There is the
side opposite  and the side adjacent to .
Hypotenuse
Side opposite 

Side adjacent to 
Copyright © 2009 Pearson Education, Inc.
Slide 6.1 - 4
Trigonometric Ratios
The lengths of the sides of a right triangle are used to
define the six trigonometric ratios:
sine (sin)
cosine (cos)
tangent (tan)
cosecant (csc)
secant (sec)
cotangent (cot)
Hypotenuse
Side opposite 

Side adjacent to 
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Slide 6.1 - 5
Trigonometric Function Values of an
Acute Angle 
Let  be an acute angle of a right triangle. Then the six
trigonometric functions of  are as follows:
side opposite 
sin 
hypotenuse
hypotenuse
csc 
side opposite 
side adjacent to 
cos 
hypotenuse
hypotenuse
sec 
side adjacent to 
side opposite 
tan 
side adjacent to 
side adjacent to 
cot  
side opposite 
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Slide 6.1 - 6
Example
In the triangle shown, find the six trigonometric

function values of (a)  and (b) .
12
13
Solution:
opp 12
a) sin 


hyp 13
5
adj 5
cos 

hyp 13
hyp 13
sec 

adj 5
opp 12
tan 

adj
5
adj
5
cot  

opp 12
hyp 13
csc 

opp 12
Copyright © 2009 Pearson Education, Inc.
Slide 6.1 - 7
Example
In the triangle shown, find the six trigonometric

function values of (a)  and (b) .
12
13
Solution:
opp 5
a) sin  


hyp 13
5
adj 12
cos  

hyp 13
hyp 13
sec  

adj 12
opp 5
tan  

adj 12
adj 12
cot  

opp 5
hyp 13
csc  

opp 5
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Slide 6.1 - 8
Reciprocal Functions
Note that there is a reciprocal relationship between
pairs of the trigonometric functions.
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
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Slide 6.1 - 9
Example
4
3
4
Given that sin   , cos   , and tan   ,
5
5
3
find csc , sec , and cot .
Solution:
1

4
5
5

4
1
1
sec  

3
cos 
5
5

3
1
csc  
sin 
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1
cot  
tan 
1
3


4
4
3
Slide 6.1 - 10
Example
6
If sin  
and  is an acute angle, find the other
7
five trigonometric function values of .
Solution:
Use the definition of the sine function that the ratio
6 opp

and draw a right triangle.
7 hyp
Use the Pythagorean equation to find a.
7
6

a
a2  b2  c2
a2  62  72
a2  36  49
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a2  49  36  13
a  13
Slide 6.1 - 11
Example
Solution continued
Use the lengths of the three sides to find the other five
ratios.
6
sin  
7
7
csc  
6
13
cos  
7
7
7 13
sec  

13
13
6
6 13
tan  

13
13
13
cot  
6
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Slide 6.1 - 12
Function Values of 45º
A right triangle with one 45º, must have a second 45º,
making it an isosceles triangle, with legs the same
length. Consider one with legs of length 1.
opp
1
2
sin 45º 


 0.7071
hyp
2
2
45º
2
1
45º
1
adj
1
2
cos 45º 


 0.7071
hyp
2
2
opp 1
tan 45º 
 1
adj 1
Copyright © 2009 Pearson Education, Inc.
Slide 6.1 - 13
Function Values of 30º
A right triangle with 30º and 60º acute angles is half an
equilateral triangle. Consider an equilateral triangle
with sides 2 and take half of it.
1
sin 30º   0.5,
2
2 30º 3
60º
1
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3
cos 30º 
 0.8660,
2
1
3
tan 30º 

 0.5774
3
3
Slide 6.1 - 14
Function Values of 60º
A right triangle with 30º and 60º acute angles is half an
equilateral triangle. Consider an equilateral triangle
with sides 2 and take half of it.
3
sin 60º 
 0.8660,
2
2 30º 3
60º
1
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1
cos 60º   0.5,
2
3
tan 60º 
 3  1.7321
1
Slide 6.1 - 15
Example
As a hot-air balloon began to rise, the ground crew
drove 1.2 mi to an observation station. The initial
observation from the station estimated the angle
between the ground and the line of sight to the balloon
to be 30º. Approximately how high was the balloon at
that point? (We are assuming that the wind velocity
was low and that the balloon rose vertically for the first
few minutes.)
Solution:
Draw the situation, label the acute angle and length of
the adjacent side.
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Slide 6.1 - 16
Example
Solution continued:
opp
h
tan 30º 

adj 1.2
 3
1.2 
h
 3 
0.7  h
1.2 tan 30º  h
The balloon is approximately 0.7 mi, or 3696 ft, high.
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Slide 6.1 - 17
Function Values of Any Acute Angle
Angles are measured either in degrees, minutes, and
seconds: 1º = 60´, 1´ = 60´´; referred to as the
DºM´S´´ form
61 degrees, 27 minutes, 42 seconds  61º 2742
or are measured in decimal degree form, expressing
the fraction parts of degrees in decimal form
61º 2742  61.45 1
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Slide 6.1 - 18
Examples
Find the trigonometric function value, rounded to four
decimal places, of each of the following:
a) tan29.7º
b) sec 48º
c) sin 84º1039
Solution:
Check that the calculator is in degree mode.
a) tan29.7º  0.5703899297  0.5704
1
b) sec 48º 
 1.49447655  1.49445
cos 48º
c) sin 84º1039  0.9948409474  0.9948
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Slide 6.1 - 19
Example
A paint crew has purchased new 30-ft extension
ladders. The manufacturer states that the safest
placement on a wall is to extend the ladder to 25 ft and
to position the base 6.5 ft from the wall. What angle
does the ladder make with the ground in this position?
Solution:
Draw the situation, label the hypotenuse and length of
the side adjacent to .
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Slide 6.1 - 20
Example
Solution continued:
adj
cos 
hyp
6.5 ft

25 ft
 0.26
Use a calculator to find the acute
angle whose cosine is 0.26:
  74.92993786º
Thus when the ladder is in its safest position, it makes
an angle of about 75º with the ground.
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Slide 6.1 - 21
Cofunction Identities
Two angles are complementary whenever the sum of
their measures is 90º. Here are some relationships.
sin  cos 90º  
90º – 

cos  sin 90º  
tan  cot 90º  
cot   tan 90º  
sec  csc 90º  
csc  sec 90º  
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Slide 6.1 - 22
Example
Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan
18º ≈ 0.3249, find the six trigonometric function values
of 72º.
Solution:
1
csc18º 
 3.2361
sin18º
sin 72º  cos18º  0.9511
cos 72º  sin18º  0.3090
1
sec18º 
 1.0515
cos18º
tan 72º  cot18º  3.0777
1
cot18º 
 3.0777
tan18º
sec 72º  csc18º  3.2361
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cot 72º  tan18º  0.3249
csc 72º  sec18º  1.0515
Slide 6.1 - 23