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The secant graph has vertical asymptotes at each value of x where the cosine graph crosses the x -axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.

Note that, because cosine is an even function, secant is also an even function. That is, sec ( x ) = sec x .

A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi/2, -pi/2, pi/2, and 3pi/2.
Graph of the secant function, f ( x ) = sec x = 1 cos x

As we did for the tangent function, we will again refer to the constant | A | as the stretching factor, not the amplitude.

Features of the graph of y = A Sec( Bx )

  • The stretching factor is | A | .
  • The period is 2 π | B | .
  • The domain is x π 2 | B | k , where k is an odd integer.
  • The range is ( , | A | ] [ | A | , ) .
  • The vertical asymptotes occur at x = π 2 | B | k , where k is an odd integer.
  • There is no amplitude.
  • y = A sec ( B x ) is an even function because cosine is an even function.

Similar to the secant, the cosecant    is defined by the reciprocal identity csc x = 1 sin x . Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0 , π , etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.

We can graph y = csc x by observing the graph of the sine function because these two functions are reciprocals of one another. See [link] . The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.

The cosecant graph has vertical asymptotes at each value of x where the sine graph crosses the x -axis; we show these in the graph below with dashed vertical lines.

Note that, since sine is an odd function, the cosecant function is also an odd function. That is, csc ( x ) = −csc x .

The graph of cosecant, which is shown in [link] , is similar to the graph of secant.

A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.
The graph of the cosecant function, f ( x ) = csc x = 1 sin x

Features of the graph of y = A Csc( Bx )

  • The stretching factor is | A | .
  • The period is 2 π | B | .
  • The domain is x π | B | k , where k is an integer.
  • The range is ( , | A | ] [ | A | , ) .
  • The asymptotes occur at x = π | B | k , where k is an integer.
  • y = A csc ( B x ) is an odd function because sine is an odd function.

Graphing variations of y = sec x And y = csc x

For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.

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?
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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.
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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?
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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
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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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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