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:

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

Questions & Answers

The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
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
Jhon Reply
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
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply

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