Secant, Cosecant, and
Cotangent Functions
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
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Printed: August 25, 2014
AUTHORS
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
www.ck12.org
C HAPTER
Chapter 1. Secant, Cosecant, and Cotangent Functions
1
Secant, Cosecant, and
Cotangent Functions
Here you’ll learn the definition of secant, cosecant, and cotangent functions and how to apply them.
While working to paint your grandfather’s staircase you are looking at the triangular shape made by the wall that
support the stairs. The staircase looks like this:
You are thinking about all of the possible relationships between sides. You already know that there are three common
relationships, called sine, cosine, and tangent.
How many others can you find?
By the end of this Concept, you’ll know the other important relationships between sides of a triangle.
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/53150
James Sousa: Graphing Cosecant and secant
Guidance
We can define three more functions also based on a right triangle. They are the reciprocals of sine, cosine and
tangent.
1
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If sin A = ac , then the definition of cosecant, or csc, is csc A = ac .
If cos A = bc , then the definition of secant, or sec, is sec A = bc .
If tan A = ab , then the definition of cotangent, or cot, is cot A = ab .
Example A
Find the secant, cosecant, and cotangent of angle B.
Solution:
First, we must find the length of the hypotenuse. We can do this using the Pythagorean Theorem:
52 + 122 = H 2
25 + 144 = H 2
169 = H 2
H = 13
Now we can find the secant, cosecant, and cotangent of angle B:
2
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Chapter 1. Secant, Cosecant, and Cotangent Functions
hypotenuse
13
=
adjacent side 12
13
hypotenuse
=
csc B =
opposite side
5
adjacent side 12
=
cot B =
opposite side
5
sec B =
Example B
Find the secant, cosecant, and cotangent of angle A
Solution:
hypotenuse
41
=
adjacent side 40
hypotenuse
41
csc A =
=
opposite side
9
adjacent side 40
cot A =
=
opposite side
9
sec A =
Example C
Find the sine, cosine, and tangent of angle A, and then use this to construct the secant, cosecant, and cotangent of
the angle
opposite side
7
=
hypotenuse
25
adjacent side 24
cos A =
=
hypotenuse
25
opposite side
7
tan A =
=
adjacent side 24
sin A =
3
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Since we know that cosecant is the reciprocal of sine, secant is the reciprocal of sine, and cotangent is the reciprocal
of tangent, we can construct these functions as follows:
1
25
=
cos A 24
25
1
=
csc A =
sin A
7
24
1
=
cot A =
tan A
7
sec A =
Vocabulary
Cosecant: The cosecant of an angle in a right triangle is a relationship found by dividing the length of the hypotenuse
by the length of the side opposite to the given angle. This is the reciprocal of the sine function.
Secant: The secant of an angle in a right triangle is a relationship found by dividing length of the hypotenuse by the
length of the side adjacent the given angle. This is the reciprocal of the cosine function.
Cotangent: The cotangent of an angle in a right triangle is a relationship found by dividing the length of the side
adjacent to the given angle by the length of the side opposite to the given angle. This is the reciprocal of the tangent
function.
Guided Practice
Find the
1. secant
2. cosecant
3. cotangent
of 6 A
Solutions:
1. The secant function is defined to be
sec =
hypotenuse
ad jacent
=
37
12
1
cos .
hypotenuse
opposite
=
37
35
1
sin .
4
ad jacent
opposite
=
12
35
sec =
hypotenuse
ad jacent .
Since sin =
opposite
hypotenuse ,
csc =
hypotenuse
opposite .
opposite
ad jacent ,
cot =
ad jacent
opposite .
≈ 1.06
3. The cotangent function is defined to be
cot =
ad jacent
hypotenuse ,
≈ 3.08
2. The cosecant function is defined to be
csc =
Since cos =
≈ .34
1
tan .
Since tan =
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Chapter 1. Secant, Cosecant, and Cotangent Functions
Concept Problem Solution
Looking at a triangle-like the shape of the wall supporting your grandfather’s staircase:
We can see that there are several ways to make relationships between the sides. In this case, we are only interested
in ratios between the sides, which means one side will be divided by another. We’ve already seen some functions,
such as:
1) The side opposite the angle divided by the hypotenuse (the sine function)
2) The side adjacent the angle divided by the hypotenuse (the cosine function)
3) The side opposite the angle divided by adjacent side (the tangent function)
In this section we introduced the reciprocal of the above trig functions. These are found by taking ratios between the
same sides shown above, except reversing the numerator and denominator:
4) The hypotenuse divided by the side opposite the angle (the cosecant function)
5) The hypotenuse divided by the side adjacent to the angle (the secant function)
6) The adjacent side divided by the opposite side (the cotangent function)
Practice
Use the diagram below for questions 1-3.
1. Find csc A and cscC.
5
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2. Find sec A and secC.
3. Find cot A and cotC.
Use the diagram to fill in the blanks below.
4.
5.
6.
7.
8.
9.
cot A = ??
cscC = ??
cotC = ??
secC = ??
csc A = ??
sec A = ??
From questions 4-9, we can conclude the following. Fill in the blanks.
10.
11.
12.
13.
14.
15.
16.
6
sec
= csc A and csc
= sec A.
cot A and cotC are _________ of each other.
Explain why the csc of an angle will always be greater than 1.
Use your knowledge of 45-45-90 triangles to find the cosecant, secant, and cotangent of a 45 degree angle.
Use your knowledge of 30-60-90 triangles to find the cosecant, secant, and cotangent of a 30 degree angle.
Use your knowledge of 30-60-90 triangles to find the cosecant, secant, and cotangent of a 60 degree angle.
As the degree of an angle increases, will the cotangent of the angle increase or decrease? Explain.