<< Chapter < Page Chapter >> Page >

Use reference angles to find all six trigonometric functions of 7 π 4 .

sin ( 7 π 4 ) = 2 2 , cos ( 7 π 4 ) = 2 2 , tan ( 7 π 4 ) = 1 , sec ( 7 π 4 ) = 2 , csc ( 7 π 4 ) = 2 , cot ( 7 π 4 ) = 1

Got questions? Get instant answers now!

Using even and odd trigonometric functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function f ( x ) = x 2 , shown in [link] . The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: ( 4 ) 2 = ( −4 ) 2 , ( −5 ) 2 = ( 5 ) 2 , and so on. So f ( x ) = x 2 is an even function, a function such that two inputs that are opposites have the same output. That means f ( x ) = f ( x ) .

This is an image of a graph of and upward facing parabola with points (-2, 4) and (2, 4) labeled.
The function f ( x ) = x 2 is an even function.

Now consider the function f ( x ) = x 3 , shown in [link] . The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So f ( x ) = x 3 is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f ( x ) = f ( x ) .

This is an image of a graph of the function f of x = x to the third power with labels for points (-1, -1) and (1, 1).
The function f ( x ) = x 3 is an odd function.

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in [link] . The sine of the positive angle is y . The sine of the negative angle is y . The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in [link] .

Graph of circle with angle of t and -t inscribed. Point of (x, y) is at intersection of terminal side of angle t and edge of circle. Point of (x, -y) is at intersection of terminal side of angle -t and edge of circle.
sin  t = y sin ( t ) = y sin  t sin ( t ) cos  t = x cos ( t ) = x cos  t = cos ( t ) tan ( t ) = y x tan ( t ) = y x tan  t tan ( t )
sec  t = 1 x sec ( t ) = 1 x sec  t = sec ( t ) csc  t = 1 y csc ( t ) = 1 y csc  t csc ( t ) cot  t = x y cot ( t ) = x y cot  t cot ( t )

Even and odd trigonometric functions

An even function is one in which f ( x ) = f ( x ) .

An odd function is one in which f ( x ) = f ( x ) .

Cosine and secant are even:

cos ( t ) = cos  t sec ( t ) = sec  t

Sine, tangent, cosecant, and cotangent are odd:

sin ( t ) = sin  t tan ( t ) = tan  t csc ( t ) = csc  t cot ( t ) = cot  t

Using even and odd properties of trigonometric functions

If the secant of angle t is 2, what is the secant of t ?

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of t is also 2.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

If the cotangent of angle t is 3 , what is the cotangent of t ?

3

Got questions? Get instant answers now!

Recognizing and using fundamental identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity    we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

Fundamental identities

We can derive some useful identities    from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

tan t = sin t cos t
sec t = 1 cos t
csc t = 1 sin t
cot t = 1 tan t = cos t sin t

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask