# 5.3 The other trigonometric functions

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In this section, you will:
• Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of $\text{\hspace{0.17em}}\frac{\pi }{3},\text{\hspace{0.17em}}$ $\frac{\pi }{4},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{\pi }{6}.$
• Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent.
• Use properties of even and odd trigonometric functions.
• Recognize and use fundamental identities.
• Evaluate trigonometric functions with a calculator.

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is $\text{\hspace{0.17em}}\frac{1}{12}\text{\hspace{0.17em}}$ or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.

## Finding exact values of the trigonometric functions secant, cosecant, tangent, and cotangent

To define the remaining functions, we will once again draw a unit circle with a point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ corresponding to an angle of $\text{\hspace{0.17em}}t,$ as shown in [link] . As with the sine and cosine, we can use the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates to find the other functions.

The first function we will define is the tangent. The tangent    of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle. In [link] , the tangent of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}\frac{y}{x},x\ne 0.\text{\hspace{0.17em}}$ Because the y -value is equal to the sine of $\text{\hspace{0.17em}}t,$ and the x -value is equal to the cosine of $\text{\hspace{0.17em}}t,$ the tangent of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ can also be defined as $\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t},\mathrm{cos}\text{\hspace{0.17em}}t\ne 0.$ The tangent function is abbreviated as $\text{\hspace{0.17em}}\text{tan}\text{.}\text{\hspace{0.17em}}$ The remaining three functions can all be expressed as reciprocals of functions we have already defined.

• The secant    function is the reciprocal of the cosine function. In [link] , the secant of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}t}=\frac{1}{x},x\ne 0.\text{\hspace{0.17em}}$ The secant function is abbreviated as $\text{\hspace{0.17em}}\text{sec}\text{.}\text{\hspace{0.17em}}$
• The cotangent    function is the reciprocal of the tangent function. In [link] , the cotangent of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}\frac{\mathrm{cos}\text{\hspace{0.17em}}t}{\mathrm{sin}\text{\hspace{0.17em}}t}=\frac{x}{y},\text{\hspace{0.17em}}y\ne 0.\text{\hspace{0.17em}}$ The cotangent function is abbreviated as $\text{\hspace{0.17em}}\text{cot}\text{.}\text{\hspace{0.17em}}$
• The cosecant    function is the reciprocal of the sine function. In [link] , the cosecant of angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}t}=\frac{1}{y},y\ne 0.\text{\hspace{0.17em}}$ The cosecant function is abbreviated as $\text{\hspace{0.17em}}\text{csc}\text{.}\text{\hspace{0.17em}}$

## Tangent, secant, cosecant, and cotangent functions

If $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is a real number and $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ is a point where the terminal side of an angle of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ radians intercepts the unit circle, then

$\begin{array}{l}\mathrm{tan}\text{\hspace{0.17em}}t=\frac{y}{x},x\ne 0\\ \mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{x},x\ne 0\\ \mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{y},y\ne 0\\ \mathrm{cot}\text{\hspace{0.17em}}t=\frac{x}{y},y\ne 0\end{array}$

## Finding trigonometric functions from a point on the unit circle

The point $\text{\hspace{0.17em}}\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)\text{\hspace{0.17em}}$ is on the unit circle, as shown in [link] . Find $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t,\mathrm{cos}\text{\hspace{0.17em}}t,\mathrm{tan}\text{\hspace{0.17em}}t,\mathrm{sec}\text{\hspace{0.17em}}t,\mathrm{csc}\text{\hspace{0.17em}}t,$ and $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t.$

Because we know the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates of the point on the unit circle indicated by angle $\text{\hspace{0.17em}}t,$ we can use those coordinates to find the six functions:

$\begin{array}{l}\mathrm{sin}\text{\hspace{0.17em}}t=y=\frac{1}{2}\\ \mathrm{cos}\text{\hspace{0.17em}}t=x=-\frac{\sqrt{3}}{2}\\ \mathrm{tan}\text{\hspace{0.17em}}t=\frac{y}{x}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{2}\left(-\frac{2}{\sqrt{3}}\right)=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\\ \mathrm{sec}\text{\hspace{0.17em}}t=\frac{1}{x}=\frac{1}{\frac{-\frac{\sqrt{3}}{2}}{}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\\ \mathrm{csc}\text{\hspace{0.17em}}t=\frac{1}{y}=\frac{1}{\frac{1}{2}}=2\\ \mathrm{cot}\text{\hspace{0.17em}}t=\frac{x}{y}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\frac{\sqrt{3}}{2}\left(\frac{2}{1}\right)=-\sqrt{3}\end{array}$

x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake