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

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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 .

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

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

The period is the same: 2 π .

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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.

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

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

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h ( x ) = 2 sec ( π 4 ( x + 1 ) )

period: 8; horizontal shift: 1 unit to left

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m ( x ) = 6 csc ( π 3 x + π )

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

1.5

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

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

5

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

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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

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

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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

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h ( x ) = 2 sec ( π 4 ( x + 1 ) )

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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

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j ( x ) = tan ( π 2 x )

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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

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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

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f ( x ) = π tan ( π x π ) π

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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

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f ( x ) = 1 4 csc ( x )

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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

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f ( x ) = 3 cot ( 2 x )

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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

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f ( x ) = 9 10 csc ( π x )

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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

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f ( x ) = sec ( x π 3 ) 2

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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

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f ( x ) = 5 ( cot ( x + π 2 ) 3 )

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

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

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 )

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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 ) sec ( x )

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

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f ( x ) = sec ( 0.001 x )

A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.
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f ( x ) = cot ( 100 π x )

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

A graph of y=1.
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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.
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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
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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.
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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.
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Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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