<< Chapter < Page Chapter >> Page >

The line y = 3 5 x passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the positive x -axis?

Got questions? Get instant answers now!

The line y = 3 7 x passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the negative x -axis?

0.405 radians

Got questions? Get instant answers now!

What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example a 5% grade means that the road rises 5 feet for every 100 feet of horizontal distance.)

Got questions? Get instant answers now!

A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder's angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

No. The angle the ladder makes with the horizontal is 60 degrees.

Got questions? Get instant answers now!

Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?

Got questions? Get instant answers now!

Chapter review exercises

Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

f ( x ) = 3 cos x + 3

amplitude: 3; period: 2 π ; midline: y = 3 ; no asymptotes

A graph of two periods of a function with a cosine parent function. The graph has a range of [0,6] graphed over -2pi to 2pi. Maximums as -pi and pi.
Got questions? Get instant answers now!

f ( x ) = 3 cos ( x + π 6 )

amplitude: 3; period: 2 π ; midline: y = 0 ; no asymptotes

A graph of four periods of a function with a cosine parent function. Graphed from -4pi to 4pi. Range is [-3,3].
Got questions? Get instant answers now!

f ( x ) = 2 sin ( x 2 π 3 )

Got questions? Get instant answers now!

f ( x ) = 3 sin ( x π 4 ) 4

amplitude: 3; period: 2 π ; midline: y = 4 ; no asymptotes

A graph of two periods of a sinusoidal function. Range is [-7,-1]. Maximums at -5pi/4 and 3pi/4.
Got questions? Get instant answers now!

f ( x ) = 2 ( cos ( x 4 π 3 ) + 1 )

Got questions? Get instant answers now!

f ( x ) = 6 sin ( 3 x π 6 ) 1

amplitude: 6; period: 2 π 3 ; midline: y = 1 ; no asymptotes

A sinusoidal graph over two periods. Range is [-7,5], amplitude is 6, and period is 2pi/3.
Got questions? Get instant answers now!

f ( x ) = 100 sin ( 50 x 20 )

Got questions? Get instant answers now!

Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

f ( x ) = tan x 4

stretching factor: none; period:   π ;   midline:   y = 4 ;   asymptotes:   x = π 2 + π k , where   k   is an integer

A graph of a tangent function over two periods. Graphed from -pi to pi, with asymptotes at -pi/2 and pi/2.
Got questions? Get instant answers now!

f ( x ) = 2 tan ( x π 6 )

Got questions? Get instant answers now!

f ( x ) = 3 tan ( 4 x ) 2

stretching factor: 3; period:   π 4 ;   midline:   y = 2 ;   asymptotes: x = π 8 + π 4 k , where   k   is an integer

A graph of a tangent function over two periods. Asymptotes at -pi/8 and pi/8. Period of pi/4. Midline at y=-2.
Got questions? Get instant answers now!

f ( x ) = 0.2 cos ( 0.1 x ) + 0.3

Got questions? Get instant answers now!

For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

f ( x ) = 1 3 sec x

amplitude: none; period: 2 π ; no phase shift; asymptotes:   x = π 2 k , where   k   is an odd integer

A graph of two periods of a secant function. Period of 2 pi, graphed from -2pi to 2pi. Asymptotes at -3pi/2, -pi/2, pi/2, and 3pi/2.
Got questions? Get instant answers now!

f ( x ) = 4 csc ( 5 x )

amplitude: none; period:   2 π 5 ;   no phase shift; asymptotes:   x = π 5 k , where   k   is an integer

A graph of a cosecant functionover two and a half periods. Graphed from -pi to pi, period of 2pi/5.
Got questions? Get instant answers now!

f ( x ) = 8 sec ( 1 4 x )

Got questions? Get instant answers now!

f ( x ) = 2 3 csc ( 1 2 x )

amplitude: none; period:   4 π ;   no phase shift; asymptotes:   x = 2 π k , where   k   is an integer

A graph of two periods of a cosecant function. Graphed from -4pi to 4pi. Asymptotes at multiples of 2pi. Period of 4pi.
Got questions? Get instant answers now!

f ( x ) = csc ( 2 x + π )

Got questions? Get instant answers now!

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y = 12 , 000 + 8 , 000 sin ( 0.628 x ), where the domain is the years since 1980 and the range is the population of the city.

What is the largest and smallest population the city may have?

largest: 20,000; smallest: 4,000

Got questions? Get instant answers now!

Graph the function on the domain of [ 0 , 40 ] .

Got questions? Get instant answers now!

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
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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