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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

The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen 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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