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Section 6.1
Trigonometric
Functions of Acute
Angles
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives

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.
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 adjacent to 
Side opposite 
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 adjacent to 
Side opposite 
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 
Example
In the triangle shown, find the six trigonometric function
values of (a)  and (b) .

opp 12
a) sin 

12
13
hyp 13
hyp 13

csc 

opp 12
5
adj
5
cos 

hyp 13
hyp 13
sec 

adj
5
opp 12
tan 

adj
5
adj
5
cot  

opp 12
Example
In the triangle shown, find the six trigonometric function
values of (a)  and (b) .

opp 5
hyp 13
a) sin  

12
13
csc  

hyp 13
opp 5

adj 12
cos  

5
hyp 13
hyp 13
sec  

adj 12
opp
5
tan  

adj 12
adj 12
cot  

opp
5
Reciprocal Functions
Note that there is a reciprocal relationship between pairs
of the trigonometric functions.
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Example
4
3
4
Given that sin   , cos   , and tan   ,
5
5
3
find csc , sec , and cot .
Solution:
csc 
sec 
1

4
5
5

4
1
1


3
cos 
5
5

3
1

sin 
1
cot  
tan 
1
3


4
4
3
Example
6
If sin   and  is an acute angle, find the other five
7
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
2
7 a. a2  b2  c2
a
 49  36  13
6
2
2
2
a

6

7
a  13

a
a2  36  49
Example (cont)
Use the lengths of the three sides to find the other five
ratios.
7
6
csc  
sin  
6
7
13
cos  
7
6
6 13
tan  

13
13
7
7 13
sec  

13
13
13
cot  
6
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
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
60º
1
3
3
cos 30º 
 0.8660,
2
1
3
tan 30º 

 0.5774
3
3
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
60º
1
3
1
cos 60º   0.5,
2
3
tan 60º 
 3  1.7321
1
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.
Example (cont)
opp
h
tan 30º 

adj 1.2
1.2 tan 30º  h
 3
1.2 
h
 3 
0.7  h
The balloon is approximately 0.7 mi, or 3696 ft, high.
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
Examples
Find the trigonometric function value, rounded to four
decimal places, of each of the following:
a) tan 29.7º
b) sec 48º
c) sin 84º1039
Solution:
Check that the calculator is in degree mode.
a) tan 29.7º  0.5703899297  0.5704
1
b) sec 48º 
 1.49447655  1.49445
cos 48º
c) sin 84º1039  0.9948409474  0.9948
Example
A window-washing 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 .
Example (cont)
6.5 ft
adj

cos 
25 ft
hyp
 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.
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  
Example
Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan
18º ≈ 0.3249, find the six trigonometric function values of
72º.
sin 72º  cos18º  0.9511
Solution:
cos 72º  sin18º  0.3090
1
csc18º 
 3.2361
sin18º
tan 72º  cot18º  3.0777
1
sec18º 
 1.0515
cot 72º  tan18º  0.3249
cos18º
1
cot18º 
 3.0777
tan18º
sec 72º  csc18º  3.2361
csc 72º  sec18º  1.0515