# 7.4 The other trigonometric functions

 Page 1 / 14
In this section you will:
• Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of $\text{\hspace{0.17em}}\frac{\pi }{3},\frac{\pi }{4},$ and $\text{\hspace{0.17em}}\frac{\pi }{6}.$
• Use reference angles to evaluate the trigonometric functions secant, 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

We can also define the remaining functions in terms of the 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 $\text{\hspace{0.17em}}\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t},\mathrm{cos}\text{\hspace{0.17em}}t\ne 0.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\text{sec}\text{.}$
• 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 $\text{\hspace{0.17em}}\frac{\mathrm{cos}\text{\hspace{0.17em}}t}{\mathrm{sin}\text{\hspace{0.17em}}t}=\frac{x}{y},y\ne 0.\text{\hspace{0.17em}}$ The cotangent function is abbreviated as $\text{\hspace{0.17em}}\text{cot}\text{.}$
• 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 $\text{\hspace{0.17em}}\text{csc}\text{.}$

## 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

## 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:

can you solve it step b step
what is linear equation with one unknown 2x+5=3
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2