Download Lesson 6 - University of Toledo

LESSON 6 THE SIX TRIGONOMETRIC FUNCTIONS
IN TERMS OF A RIGHT TRIANGLE
Topics in this lesson:
1.
DEFINITION AND EXAMPLES OF THE TRIGONOMETRIC
FUNCTIONS OF AN ACUTE ANGLE IN TERMS OF A RIGHT
TRIANGLE
2.
USING A RIGHT TRIANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS OF ANGLES IN THE FIRST, SECOND,
THIRD, AND FOURTH QUADRANTS
1.
DEFINITION AND EXAMPLES OF THE TRIGONOMETRIC
FUNCTIONS OF AN ACUTE ANGLE IN TERMS OF A RIGHT
TRIANGLE

hypotenuse
hypotenuse
opposite side
of 
adjacent side
of 

adjacent side
of 
opposite side
of 
Definition Given the angle  in the triangle above. We define the following
cos  
adj
hyp
sec  
hyp
adj
sin  
opp
hyp
csc  
hyp
opp
tan  
opp
adj
cot  
adj
opp
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Illustration of the definition of the cosine function, sine function, tangent function,
secant function, cosecant function, and cotangent function for the acute angle 
using right triangle trigonometry.
Second illustration of the cosine function, sine function, tangent function, secant
function, cosecant function, and cotangent function for the acute angle  using
right triangle trigonometry.
Illustration of the definition of all the six trigonometric functions for an acute angle
 using right triangle trigonometry. Second illustration of all the six trigonometric
functions.
NOTE: Since the three angles of any triangle sum to 180  and the right angle in
the triangle is 90  , then the other two angles in the right triangle must sum to 90  .
Thus, the other two angles in the triangle must be greater than 0  and less than
90  . Thus, the other two angles in the triangle are acute angles. Thus, the angle 
above is an acute angle. If we consider the angle  in standard position, then  is
in the first quadrant and we would have the following:
y
r
Pr ( )  ( x , y )
hypotenuse = r
y = opposite side of 

-r
x = adjacent
side of 
r
x
-r
cos  
x
adj

r
hyp
sec  
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
r
hyp

x
adj
sin  
y
opp

r
hyp
csc  
r
hyp

y
opp
tan  
y
opp

x
adj
cot  
x
adj

y
opp
The advantage to this definition is that the angle  does not have to be in standard
position in order to recognize the hypotenuse of the triangle and the opposite and
adjacent side of the angle  . Thus, the right triangle can be oriented anyway in the
plane. The triangle could be spun in the plane and when it stopped spinning, you
would still be able to identify the hypotenuse of the triangle and the opposite and
adjacent side of the angle  .
One disadvantage of this definition is that the angle  must be an acute angle. This
would exclude any angle whose terminal side lies on one of the coordinate axes. It
would also exclude any angle whose terminal side lies in the second, third, fourth
and first (by rotating clockwise) quadrants; however, the reference angle for these
angles would be acute and could be put into a right triangle.
Examples Find the exact value of the six trigonometric functions for the following
angles.
1.
5

2
Using the Pythagorean Theorem to find the length of the hypotenuse, we
have that the length of the hypotenuse is 4  25  29 . Thus, we have
that
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
29
5 = opposite side of 

2 = adjacent side of 
cos  
adj

hyp
2
sin  
opp

hyp
5
tan  
opp
5

adj
2
29
29
sec  
hyp

adj
csc  
hyp

opp
cot  
adj
2

opp
5
29
2
29
5
2.

8
4
Using the Pythagorean Theorem to find the length of the second side, we have that
the length of the second side is 64  16  48  4 3 . Thus, we have that
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330

8
adjacent side of  = 4 3
4 = opposite side of 
cos  
4 3
3
adj


hyp
8
2
sec  
sin  
opp
4
1


hyp
8
2
csc   2
tan  
opp
4
1


adj
4 3
3
cot  
2
3
3
NOTE: These answers should look familiar to you. The angle  would
have to be the 30  or

angle.
6
Examples Use a right triangle to find the exact value of the other five
trigonometric functions if given the following.
1.
sin  
sin  
3
7
3
7
and  is an acute angle

opp
hyp
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
7
3

46
NOTE:
The number circled above was found using the Pythagorean
Theorem by 49  3  46 .
cos  
sin  
tan  
2.
adj

hyp
46
7
3
7
(given)
opp

adj
3
46

3
46
sec  
7
csc  
7
cot  
tan   6 and  is an acute angle
tan   6 
6
opp

1
adj
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
46
3
46
3

46
3
37
6

1
NOTE:
The number circled above was found using the Pythagorean
Theorem by 1  36  37 .
cos  
adj

hyp
1
sin  
opp

hyp
6
37
37
sec  
12
and  is an acute angle
5
sec  
12

5
cos  
37
csc  
37
cot  
tan   6 (given)
3.
sec  
5
adj

12
hyp
12
119

5
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
6
1
6
NOTE:
The number circled above was found using the Pythagorean
Theorem by 144  25  119 .
cos  
5
12
sin  
opp

hyp
tan  
opp

adj
sec  
119
12
119
5
12
(given)
5
csc  
12
cot  
5
119
119
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2.
USING A RIGHT TRIANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS OF ANGLES IN THE FIRST,
SECOND, THIRD, AND FOURTH QUADRANTS
Examples Determine what quadrant the following angles are in if given the
following information.
1.
cos   0 and sin   0
Method 1:
cos   0  the x-coordinate of the point of intersection of the terminal
side of the angle  with the Unit Circle is negative. That is, x < 0.
sin   0  the y-coordinate of the point of intersection of the terminal
side of the angle  with the Unit Circle is positive. That is, y > 0.
Thus, we have that x < 0 and y > 0. Thus, the angle  is in the II quadrant.
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Method 2:
cos   0
sin   0
y
y
X
X
x
X
x
X
Thus, the common quadrant is the II quadrant.
Method 3:
cos   0   is in either the II or III quadrant
sin   0   is in either the I or II quadrant
Thus, the common quadrant is the II quadrant.
Answer: II
2.
tan   0 and cos   0
Method 1:
cos   0  the x-coordinate of the point of intersection of the terminal
side of the angle  with the Unit Circle is positive. That is, x > 0.
The tangent of the angle  is the y-coordinate of the point of intersection of
the terminal side of the angle  with the Unit Circle divided by the xy
tan


coordinate of the point of intersection. That is,
. Since we have
x
that tan   0 and x > 0, then we have the following:
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
(  )  tan  
y
?

 y  0
x
()
since only a negative divided by a positive will result in a negative.
Thus, we have that x > 0 and y < 0. Thus, the angle  is in the IV quadrant.
Method 2:
cos   0
tan   0
y
y
X
X
x
X
X
Thus, the common quadrant is the IV quadrant.
Method 3:
tan   0   is in either the II or IV quadrant
cos   0   is in either the I or IV quadrant
Thus, the common quadrant is the IV quadrant.
Answer: IV
3.
x
sin   0 and cot   0
Method 1:
sin   0  y  0
Copyrighted by James D. Anderson, The University of Toledo
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cot   0  tan   0 . Since we have that tan   0 and y < 0, then we
have the following:
(  )  tan  
()
y

 x  0
x
?
since only a negative divided by a negative will result in a positive.
Thus, we have that x < 0 and y < 0. Thus, the angle  is in the III quadrant.
Method 2:
cot   0 or tan   0
sin   0
y
y
X
x
X
X
x
X
Thus, the common quadrant is the III quadrant.
Method 3:
sin   0   is in either the III or IV quadrant
cot   0   is in either the I or III quadrant
or tan   0   is in either the I or III quadrant
Thus, the common quadrant is the III quadrant.
Answer: III
4.
tan   0 and sec   0
Method 1:
Copyrighted by James D. Anderson, The University of Toledo
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sec   0  cos   0  x  0
Since we have that tan   0 and x < 0, then we have the following:
(  )  tan  
y
?

 y0
x
()
since only a positive divided by a negative will result in a negative.
Thus, we have that x < 0 and y > 0. Thus, the angle  is in the II quadrant.
Method 2:
sec   0 or cos   0
tan   0
y
y
X
X
x
X
x
X
Thus, the common quadrant is the II quadrant.
Method 3:
tan   0   is in either the II or IV quadrant
sec   0   is in either the II or III quadrant
or cos   0   is in either the II or III quadrant
Thus, the common quadrant is the II quadrant.
Answer: II
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
5.
csc   0 and sec   0
Method 1:
csc   0  sin   0  y  0
sec   0  cos   0  x  0
Thus, we have that x > 0 and y < 0. Thus, the angle  is in the IV quadrant.
Method 2:
csc   0 or sin   0
sec   0 or cos   0
y
y
X
x
X
X
x
X
Thus, the common quadrant is the IV quadrant.
Method 3:
csc   0   is in either the III or IV quadrant
or sin   0   is in either the III or IV quadrant
sec   0   is in either the I or IV quadrant
or cos   0   is in either the I or IV quadrant
Thus, the common quadrant is the IV quadrant.
Answer: IV
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Examples Use a right triangle to find the exact value of the other five
trigonometric functions if given the following.
1.
cos  
11
6
and  is in the IV quadrant
Since the angle  is in the IV quadrant, then it is not an acute angle. Thus,
the angle  can not be put in a right triangle. Thus, we have that
11
cos  
6

adj
hyp
However, the reference angle  ' of the angle  is an acute and can be put in
a triangle. Thus, we have that
cos  
11
 cos  ' 
6
11
6

adj
hyp
6
5
'
11
NOTE:
The number circled above was found using the Pythagorean
Theorem by 36  11  25  5 .
From the triangle, we obtain that
cos  ' 
11
6
(“given”)
sin  ' 
opp
5

hyp
6
Copyrighted by James D. Anderson, The University of Toledo
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tan  ' 
opp

adj
5
11
Since the angle  is in the IV quadrant, then we know that the sine (and
cosecant) of  is negative and the tangent (and cotangent) of  is negative.
Now, using reference angles, we obtain that
cos  
11
6
sin    sin  '  
5
6
csc   
csc   
5
tan    tan  '  
2.
sec  
(given)
cot   
11
6
11
6
5
11
5
17
and cot   0
8
First, we will use Method 1 from above in order to determine what quadrant
the terminal side of the angle  is in.
csc   0  sin   0  y  0
cot   0  tan   0 . Since we have that tan   0 and y < 0, then we
have the following:
(  )  tan  
()
y

 x  0
x
?
since only a negative divided by a negative will result in a positive.
Thus, we have that x < 0 and y < 0. Thus, the angle  is in the III quadrant.
You may use Method 2 or Method 3 from above if you prefer.
csc   
17
8
 sin   
8
17
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Since the angle  is in the III quadrant, then it is not an acute angle. Thus,
the angle  can not be put in a right triangle. Thus, we have that
sin   
8
opp

17
hyp
However, the reference angle  ' of the angle  is an acute and can be put in
a triangle. Thus, we have that
sin   
8
8
opp
 sin  ' 

17
17
hyp
17
8
'
15
NOTE:
The number circled above was found using the Pythagorean
Theorem by 289  64  225  15 .
From the triangle, we obtain that
cos  ' 
adj
15

hyp
17
sin  ' 
8
(“given”)
17
tan  ' 
opp
8

adj
15
Since the angle  is in the III quadrant, then we know that the cosine (and
secant) of  is negative and the tangent (and cotangent) of  is positive.
Now, using reference angles, we obtain that
cos    cos  '  
15
17
sec   
Copyrighted by James D. Anderson, The University of Toledo
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17
15
sin   
8
17
csc   
tan   tan  ' 
3.
tan   
8
15
cot  
17
(given)
8
15
8
5
and sec   0
7
First, we will use Method 1 from above in order to determine what quadrant
the terminal side of the angle  is in.
sec   0  cos   0  x  0
Since we have that tan   0 and x < 0, then we have the following:
(  )  tan  
y
?

 y  0
x
()
since only a positive divided by a negative will result in a negative.
Thus, we have that x < 0 and y > 0. Thus, the angle  is in the II quadrant.
You may use Method 2 or Method 3 from above if you prefer.
Since the angle  is in the II quadrant, then it is not an acute angle. Thus, the
angle  can not be put in a right triangle. Thus, we have that
tan   
5
opp

7
adj
However, the reference angle  ' of the angle  is an acute and can be put in
a triangle. Thus, we have that
tan   
5
5
opp
 tan  ' 

7
7
adj
Copyrighted by James D. Anderson, The University of Toledo
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74
5
'
7
NOTE:
The number circled above was found using the Pythagorean
Theorem by 49  25  74 .
From the triangle, we obtain that
cos  ' 
adj

hyp
7
sin  ' 
74
opp

hyp
5
74
tan  ' 
5
(“given”)
7
Since the angle  is in the II quadrant, then we know that the cosine (and
secant) of  is negative and the sine (and cosecant) of  is positive. Now,
using reference angles, we obtain that
cos    cos  '  
sin   sin  ' 
tan   
4.
sin  
7
74
5
74
5
(given)
7
74
sec   
csc  
cot   
5
and  is in the II quadrant
9
Copyrighted by James D. Anderson, The University of Toledo
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7
74
5
7
5
Since the angle  is in the II quadrant, then it is not an acute angle. Thus,
the angle  can not be put in a right triangle. Thus, we have that
5
opp

9
hyp
sin  
However, the reference angle  ' of the angle  is an acute and can be put in
a triangle. Thus, we have that
sin  
5
5
opp
 sin  ' 

9
9
hyp
9
5
'
56
NOTE:
The number circled above was found using the Pythagorean
Theorem by 81  25  56 .
From the triangle, we obtain that
cos  ' 
adj

hyp
56
9
sin  ' 
5
(“given”)
9
tan  ' 
opp

adj
5
56
Since the angle  is in the II quadrant, then we know that the cosine (and
secant) of  is negative and the tangent (and cotangent) of  is negative.
Now, using reference angles, we obtain that
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
cos    cos  '  
sin  
sec   
9
5
(given)
9
tan    tan  '  
5.
56
csc  
5
56
9
5
cot   
56
9
56
5
sec    5 and sin   0
First, we will use Method 1 from above in order to determine what quadrant
the terminal side of the angle  is in.
sec   0  cos   0  x  0
sin   0  y  0
Thus, we have that x < 0 and y < 0. Thus, the angle  is in the III quadrant.
You may use Method 2 or Method 3 from above if you prefer.
sec    5  cos   
1
5
Since the angle  is in the III quadrant, then it is not an acute angle. Thus,
the angle  can not be put in a right triangle. Thus, we have that
cos   
1
adj

5
hyp
However, the reference angle  ' of the angle  is an acute and can be put in
a triangle. Thus, we have that
Copyrighted by James D. Anderson, The University of Toledo
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cos   
1
1
adj
 cos  ' 

5
5
hyp
5
24
'
1
NOTE:
The number circled above was found using the Pythagorean
Theorem by 25  1  24 .
From the triangle, we obtain that
cos  ' 
1
(“given”)
5
sin  ' 
opp

hyp
24
5
tan  ' 
opp

adj
24
Since the angle  is in the III quadrant, then we know that the sine (and
cosecant) of  is negative and the tangent (and cotangent) of  is positive.
Now, using reference angles, we obtain that
cos   
1
5
sec    5
sin    sin  '  
tan   tan  ' 
24
24
5
5
csc   
cot  
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Copyrighted by James D. Anderson, The University of Toledo
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(given)
24
1
24