<< Chapter < Page Chapter >> Page >

cos x = 0 ; x = 2 n + 1 π 2 , where n Z

The function cos(x) is zero for x = ± π 2, ± 3 π 2, ± 5 π 2, etc. In other words, the cosine function is zero for all odd multiples of “ π / 2 ”. It means that tangent function is not defined for odd multiples of “ π / 2 ”. Therefore, values of “x” for which cosine is zero need to be excluded from the domain set of real number set “R”. On the other hand, the values of tangent function are extended along the real number line on either side of zero. The range of the function, therefore, is “R”. Hence, domain and range of tangent function are :

Domain = R { x : x = 2 n + 1 π 2 , n Z }

Range = R

The plot of tan(x) .vs. x is shown here.

Tangent function

Graph of tangent function

The period of tanx is π. Multiplication of tangent function by a constant A does not change the range as in the case of sine and cosine function. The plot is always extended on either side of x-axis so that its range is R. Multiplying argument x like tan(kx), however, changes the points where function is not defined. It is now given by :

x = 2 n + 1 π 2k , n Z }

Therefore, domain is now modified as :

Domain = R { x : x = 2 n + 1 π 2k , n Z }

Cosecant function

For a real number “x”, there is a cosecant function defined as :

f x = cosec x

Again, the function is not defined for all real number “x”. Let us recall that :

cosec x = 1 sin x

This is a rational polynomial form, which is defined for sin x 0 . Now, sin(x) evaluates to zero for values of “x”, which appears at a certain interval given by the condition,

sin x = 0 ; x = n π , where n Z

This means that sin(x) is zero for x = 0, ± π , ± 2 π , ± 3 π , etc. In other words, the sine function is zero for all integer multiples of “ π ”. It means that cosecant function is not defined for integral multiples of “ π ”. Therefore, values of “x”, for which sine is zero, need to be excluded from real number set “R” for defining domain of the function.

On the other hand, values of cosecant function are fall in certain intervals. We have seen that values of sine function is between “-1” and “1”, including end points. Reciprocal of these values are either lesser than “-1” or greater than “1”. Symbolically,

cosec x - 1

cosec x 1

Combining two intervals, using modulus function :

| cosec x | 1

The combined interval of the cosecant function, therefore, is :

( - , - 1 ] ( 1, ]

Hence, domain and range of cosecant function are :

Domain = R { x : x = n π , n Z }

Range = ( - , - 1 ] ( 1, ]

The plot of cosecant(x) .vs. x is shown here.

Cosecant function

Graph of cosecant function

The period of cosecx is 2π. Important to note here is that function is not defined even within a periodic segment. Since cosec(-x) = -cosecx, we conclude that cosecant function is odd function in each of periodic segment. Multiplication of cosecant function by a constant A changes the range as plot lies on or beyond -A or A. The range is modified as :

Range = ( - , - A ] ( A, ]

Multiplying argument x like cosec(kx), however, changes the points where function is not defined. It is now given by :

x = n π k , n Z

Therefore, domain is now modified as :

Domain = R { x : x = n π k , n Z }

Secant function

For a real number “x”, there is a secant function defined as :

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask