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How can the graph of y = cos x be used to construct the graph of y = sec x ?

Explain why the period of tan x is equal to π .

Answers will vary. Using the unit circle, one can show that tan ( x + π ) = tan x .

Why are there no intercepts on the graph of y = csc x ?

How does the period of y = csc x compare with the period of y = sin x ?

The period is the same: 2 π .

Algebraic

For the following exercises, match each trigonometric function with one of the following graphs.

Trigonometric graph of tangent of x. Trigonometric graph of secant of x. Trigonometric graph of cosecant of x. Trigonometric graph of cotangent of x.

f ( x ) = tan x

f ( x ) = sec x

IV

f ( x ) = csc x

f ( x ) = cot x

III

For the following exercises, find the period and horizontal shift of each of the functions.

f ( x ) = 2 tan ( 4 x 32 )

h ( x ) = 2 sec ( π 4 ( x + 1 ) )

period: 8; horizontal shift: 1 unit to left

m ( x ) = 6 csc ( π 3 x + π )

If tan x = 1.5 , find tan ( x ) .

1.5

If sec x = 2 , find sec ( x ) .

If csc x = 5 , find csc ( x ) .

5

If x sin x = 2 , find ( x ) sin ( x ) .

For the following exercises, rewrite each expression such that the argument x is positive.

cot ( x ) cos ( x ) + sin ( x )

cot x cos x sin x

cos ( x ) + tan ( x ) sin ( x )

Graphical

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

f ( x ) = 2 tan ( 4 x 32 )

A graph of two periods of a modified tangent function. There are two vertical asymptotes.

stretching factor: 2; period:   π 4 ;   asymptotes:   x = 1 4 ( π 2 + π k ) + 8 ,  where  k  is an integer

h ( x ) = 2 sec ( π 4 ( x + 1 ) )

m ( x ) = 6 csc ( π 3 x + π )

A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.

stretching factor: 6; period: 6; asymptotes:   x = 3 k ,  where  k  is an integer

j ( x ) = tan ( π 2 x )

p ( x ) = tan ( x π 2 )

A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.

stretching factor: 1; period:   π ;   asymptotes:   x = π k ,  where  k  is an integer

f ( x ) = 4 tan ( x )

f ( x ) = tan ( x + π 4 )

A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.

Stretching factor: 1; period:   π ;   asymptotes:   x = π 4 + π k ,  where  k  is an integer

f ( x ) = π tan ( π x π ) π

f ( x ) = 2 csc ( x )

A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.

stretching factor: 2; period:   2 π ;   asymptotes:   x = π k ,  where  k  is an integer

f ( x ) = 1 4 csc ( x )

f ( x ) = 4 sec ( 3 x )

A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2.

stretching factor: 4; period:   2 π 3 ;   asymptotes:   x = π 6 k ,  where  k  is an odd integer

f ( x ) = 3 cot ( 2 x )

f ( x ) = 7 sec ( 5 x )

A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart.

stretching factor: 7; period:   2 π 5 ;   asymptotes:   x = π 10 k ,  where  k  is an odd integer

f ( x ) = 9 10 csc ( π x )

f ( x ) = 2 csc ( x + π 4 ) 1

A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.

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

f ( x ) = sec ( x π 3 ) 2

f ( x ) = 7 5 csc ( x π 4 )

A graph of a modified cosecant function. Four vertical asymptotes.

stretching factor:   7 5 ;   period:   2 π ;   asymptotes:   x = π 4 + π k ,  where  k  is an integer

f ( x ) = 5 ( cot ( x + π 2 ) 3 )

For the following exercises, find and graph two periods of the periodic function with the given stretching factor, | A | , period, and phase shift.

A tangent curve, A = 1 , period of π 3 ; and phase shift ( h , k ) = ( π 4 , 2 )

y = tan ( 3 ( x π 4 ) ) + 2

A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.

A tangent curve, A = −2 , period of π 4 , and phase shift ( h , k ) = ( π 4 , −2 )

For the following exercises, find an equation for the graph of each function.

A graph of two periods of a modified cosecant function, with asymptotes at multiples of pi/2.

f ( x ) = csc ( 2 x )

A graph of a modified cotangent function. Vertical asymptotes at x=-1 and x=0 and x=1.
A graph of a modified cosecant function. Vertical asymptotes at multiples of pi/4.

f ( x ) = csc ( 4 x )

A graph of a modified tangent function. Vertical asymptotes at -pi/8 and 3pi/8.
A graph of a modified cosecant function. Vertical asymptotyes at multiples of pi.

f ( x ) = 2 csc x

A graph of a modified secant function. Four vertical asymptotes.
graph of two periods of a modified tangent function. Vertical asymptotes at x=-0.005 and x=0.005.

f ( x ) = 1 2 tan ( 100 π x )

Technology

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input csc x as 1 sin x .

f ( x ) = | csc ( x ) |

f ( x ) = | cot ( x ) |

A graph of the absolute value of the cotangent function. Range is 0 to infinity.

f ( x ) = 2 csc ( x )

f ( x ) = csc ( x ) sec ( x )

A graph of tangent of x.

Graph f ( x ) = 1 + sec 2 ( x ) tan 2 ( x ) . What is the function shown in the graph?

f ( x ) = sec ( 0.001 x )

A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.

f ( x ) = cot ( 100 π x )

f ( x ) = sin 2 x + cos 2 x

A graph of y=1.

Real-world applications

The function f ( x ) = 20 tan ( π 10 x ) marks the distance in the movement of a light beam from a police car across a wall for time x , in seconds, and distance f ( x ) , in feet.

  1. Graph on the interval [ 0 , 5 ] .
  2. Find and interpret the stretching factor, period, and asymptote.
  3. Evaluate f ( 1 ) and f ( 2.5 ) and discuss the function’s values at those inputs.

Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x , measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x is measured negative to the left and positive to the right. (See [link] .) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance d ( x ) , in kilometers, from the fisherman to the boat is given by the function d ( x ) = 1.5 sec ( x ) .

  1. What is a reasonable domain for d ( x ) ?
  2. Graph d ( x ) on this domain.
  3. Find and discuss the meaning of any vertical asymptotes on the graph of d ( x ) .
  4. Calculate and interpret d ( π 3 ) . Round to the second decimal place.
  5. Calculate and interpret d ( π 6 ) . Round to the second decimal place.
  6. What is the minimum distance between the fisherman and the boat? When does this occur?
An illustration of a man and the distance he is away from a boat.
  1. ( π 2 , π 2 ) ;
  2. A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.
  3. x = π 2 and x = π 2 ; the distance grows without bound as | x | approaches π 2 —i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
  4. 3; when x = π 3 , the boat is 3 km away;
  5. 1.73; when x = π 6 , the boat is about 1.73 km away;
  6. 1.5 km; when x = 0

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 ) = 250,000 csc ( π 30 x ) .

  1. Graph g ( x ) on the interval [ 0 , 35 ] .
  2. Evaluate g ( 5 ) and interpret the information.
  3. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?
  4. Find and discuss the meaning of any vertical asymptotes.

A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after x seconds is π 120 x .

  1. Write a function expressing the altitude h ( x ) , in miles, of the rocket above the ground after x seconds. Ignore the curvature of the Earth.
  2. Graph h ( x ) on the interval ( 0 , 60 ) .
  3. Evaluate and interpret the values h ( 0 ) and h ( 30 ) .
  4. What happens to the values of h ( x ) as x approaches 60 seconds? Interpret the meaning of this in terms of the problem.
  1. h ( x ) = 2 tan ( π 120 x ) ;
  2. An exponentially increasing function with a vertical asymptote at x=60.
  3. h ( 0 ) = 0 : after 0 seconds, the rocket is 0 mi above the ground; h ( 30 ) = 2 : after 30 seconds, the rockets is 2 mi high;
  4. As x approaches 60 seconds, the values of h ( x ) grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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