1.9: Secant, Cosecant, and Cotangent Functions
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
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.
If \begin{align*}\sin A = \frac{a}{c}\end{align*}
If \begin{align*}\cos A = \frac{b}{c}\end{align*}
If \begin{align*}\tan A = \frac{a}{b}\end{align*}
Example A
Find the secant, cosecant, and cotangent of angle \begin{align*}B\end{align*}
Solution:
First, we must find the length of the hypotenuse. We can do this using the Pythagorean Theorem:
\begin{align*}5^2 + 12^2 & = H^2\\
25 + 144 & = H^2\\
169 & = H^2\\
H & = 13\end{align*}
Now we can find the secant, cosecant, and cotangent of angle \begin{align*}B\end{align*}
\begin{align*}\sec B & = \frac{\text{hypotenuse}}{\text{adjacent side}} = \frac{13}{12}\\
\csc B & = \frac{\text{hypotenuse}}{\text{opposite side}} = \frac{13}{5}\\
\cot B & = \frac{\text{adjacent side}}{\text{opposite side}} = \frac{12}{5}\end{align*}
Example B
Find the secant, cosecant, and cotangent of angle \begin{align*}A\end{align*}
Solution:
\begin{align*}\sec A & = \frac{\text{hypotenuse}}{\text{adjacent side}} = \frac{41}{40}\\
\csc A & = \frac{\text{hypotenuse}}{\text{opposite side}} = \frac{41}{9}\\
\cot A & = \frac{\text{adjacent side}}{\text{opposite side}} = \frac{40}{9}\end{align*}
Example C
Find the sine, cosine, and tangent of angle \begin{align*}A\end{align*}
\begin{align*}\sin A & = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{7}{25}\\
\cos A & = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{24}{25}\\
\tan A & = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{7}{24}\end{align*}
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:
\begin{align*}\sec A & = \frac{1}{\cos A} = \frac{25}{24}\\
\csc A & = \frac{1}{\sin A} = \frac{25}{7}\\
\cot A & = \frac{1}{\tan A} = \frac{24}{7}\end{align*}
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 \begin{align*}\angle A\end{align*}
Solutions:
1. The secant function is defined to be \begin{align*}\frac{1}{\cos}\end{align*}
\begin{align*}\sec = \frac{hypotenuse}{adjacent} = \frac{37}{12} \approx 3.08\end{align*}
2. The cosecant function is defined to be \begin{align*}\frac{1}{\sin}\end{align*}
\begin{align*}\csc = \frac{hypotenuse}{opposite} = \frac{37}{35} \approx 1.06 \end{align*}
3. The cotangent function is defined to be \begin{align*}\frac{1}{\tan}\end{align*}
\begin{align*}\cot = \frac{adjacent}{opposite} = \frac{12}{35} \approx .34 \end{align*}
Concept Problem Solution
Looking at a trianglelike 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 13.
 Find \begin{align*}\csc A\end{align*}
cscA and \begin{align*}\csc C\end{align*}cscC .  Find \begin{align*}\sec A\end{align*}
secA and \begin{align*}\sec C\end{align*}secC .  Find \begin{align*}\cot A\end{align*}
cotA and \begin{align*}\cot C\end{align*}cotC .
Use the diagram to fill in the blanks below.

\begin{align*}\cot A = \frac{?}{?}\end{align*}
cotA=?? 
\begin{align*}\csc C = \frac{?}{?}\end{align*}
cscC=?? 
\begin{align*}\cot C = \frac{?}{?}\end{align*}
cotC=?? 
\begin{align*}\sec C = \frac{?}{?}\end{align*}
secC=?? 
\begin{align*}\csc A = \frac{?}{?}\end{align*}
cscA=?? 
\begin{align*}\sec A = \frac{?}{?}\end{align*}
secA=??
From questions 49, we can conclude the following. Fill in the blanks.

\begin{align*}\sec \underline{\;\;\;\;\;\;\;} = \csc A\end{align*}
sec−−−−=cscA and \begin{align*}\csc \underline{\;\;\;\;\;\;\;} = \sec A\end{align*}csc−−−−=secA . 
\begin{align*}\cot A\end{align*}
cotA and \begin{align*}\cot C\end{align*}cotC are _________ of each other.  Explain why the csc of an angle will always be greater than 1.
 Use your knowledge of 454590 triangles to find the cosecant, secant, and cotangent of a 45 degree angle.
 Use your knowledge of 306090 triangles to find the cosecant, secant, and cotangent of a 30 degree angle.
 Use your knowledge of 306090 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.
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.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.Secant
The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.Image Attributions
Here you'll learn the definition of secant, cosecant, and cotangent functions and how to apply them.
Concept Nodes:
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.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.Secant
The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.