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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 golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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