0. Code

>	restart; with(plots):

Warning, the name changecoords has been redefined

>

SolidPlotBound := proc(f,a,b, M) local  n,delta,x1,x2,y1,y2,B,i: n:= 60;   delta := (b-a)/n;   x2 := a; for i from 1 to n do x1 := evalf(x2);          y1 := evalf( f(x1)); x2 := evalf(a + i*delta); y2 := evalf( f(x2));   if(y1 > M) then y1 := M; else if (y1 < -M) then y1 := -M; fi;fi;   if(y2 > M) then y2 := M; else if (y2 < -M) then y2 := -M;fi;fi;   B[i]:=polygonplot([[x1,0],[x1,y1],[x2,y2],[x2,0]], color=red, style=patchnogrid); od; display({ seq( B[i],i=1..n )  } ); end:

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 1. The Tangent Graph

y = tan(x) is also a function. It too is periodic. However, it differs from sine and cosine because its period is only  and it has vertical asymptotes.

>	f := x -> tan(x):

>	SolidPlotBound( x -> tan(x), -1.56, 1.56, 20);

This is one period of the tangent. The tangent is continuous on the interval ( ), and this constitutes a complete period of length . Of course, it repeats this same pattern indefinitely.

>	plot( tan(x), x = -2*Pi..2*Pi, y = -20..20, discont = true);

Notice that there are vertical asymptotes for all values of x of the form

>	display( plot( tan(x), x = -2*Pi..2*Pi, y = -20..20,                 discont = true, color = blue, thickness = 2),             seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]],                         color = green, linestyle = 2),          k = -2..1) );

>	display( plot( tan(x), x = -6*Pi..6*Pi, y = -20..20,                 discont = true, color = blue, thickness = 1),             seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]],                         color = green, linestyle = 2),          k = -6..5) );

 2. The Secant Graph

The secant is the reciprocal of the cosine. So this helps us to understand the graph. Since -1 <= cos(x) <= 1, this means secant is entirely in the intervals . Here is the cosine, and then the cosine with its reciprocal, the secant.

>	display( [SolidPlotBound( x -> cos(x), 0, 2*Pi, 5),                     plot(sec(x), x = 0..2*Pi, y = -5..5,                         color = blue, thickness = 2, discont = true )]);

Like the tangent function, the secant is undefined whenever cosine is zero. This means there are vertical asymptotes in the exact same places as the tangent function - all values of x of the form .

>	display( plot( sec(x), x = -2*Pi..2*Pi, y = -20..20,                 discont = true, color = blue, thickness = 2),             seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]],                         color = green, linestyle = 2),          k = -2..1) );

>	display( plot( sec(x), x = -6*Pi..6*Pi, y = -20..20,                 discont = true, color = blue, thickness = 1),             seq( plot( [[(2*k+1)*Pi/2 ,-20],[(2*k+1)*Pi/2 ,20]],                         color = green, linestyle = 2),          k = -6..5) );

Just like the cosine, when we alter the function in various ways, we affect its period and height. The secant doesn't have an amplitude, but its minimal positive value is the reciprocal of the amplitude of the associate cosine.

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display( [SolidPlotBound( x -> 2*cos(3*x), 0, 2*Pi, 5),                     plot( (1/2)*sec(3*x), x = 0..2*Pi, y = -5..5,                         color = blue, thickness = 2, discont = true )]); `Notice that the period is 2*Pi/3.`; `The amplitude of the cosine is 2, while the minimal positive value of the secant is 1/2`;

>	display( [SolidPlotBound( x -> (1/2)*cos((1/2)*x), 0, 4*Pi, 5),                     plot( 2*sec((1/2)*x), x = 0..4*Pi, y = -5..5,                         color = blue, thickness = 2, discont = true )]);

 3. The Cosecant Graph

The cosecant is the reciprocal of the sine, and this helps us to understand it in the same way we understood secant. Like the cosine, the sine is limited to the same range -1 <= sin(x) <= 1, this means csc is entirely in the intervals . Also, if you recall, the cosine "trails"  behind the sine, so sec will follow  behind the csc in the same way - thus the graph of csc will look the same as the graph of the sec - excepted shifted by

Here is the sine, and then the cosine with its reciprocal, the secant.

>	SolidPlotBound( x -> sin(x), 0, 2*Pi, 10);

>	display( [SolidPlotBound( x -> sin(x), 0, 2*Pi, 5),                     plot(csc(x), x = 0..2*Pi, y = -5..5,                         color = blue, thickness = 2, discont = true )]);

Like the tangent  and secant functions, the cosecant is undefined at certain values. Those two are undefined when cos(x) = 0, while csc is undefined when sin(x) = 0. This means there are vertical asymptotes but in different places - when x = kPi.

>	display( plot( csc(x), x = -2*Pi..2*Pi, y = -20..20,                 discont = true, color = blue, thickness = 2),             seq( plot( [[ k*Pi,-20],[k*Pi ,20]],                         color = green, linestyle = 2),          k = -2..1) );

>	display( plot( csc(x), x = -6*Pi..6*Pi, y = -20..20,                 discont = true, color = blue, thickness = 1),             seq( plot( [[k*Pi ,-20],[k*Pi ,20]],                         color = green, linestyle = 2),          k = -6..5) );

 4. Secants & Cosecants - Together at Last

To help see the differences in the secant and cosecant, let's look at them on the same graph.

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display( plot( sin(x), x=0..2*Pi, color = blue),          plot( cos(x), x=0..2*Pi, color = green ),          plot( csc(x), x=0..2*Pi, y = -10..10,  color = blue,                                                 thickness = 3, discont = true),          plot( sec(x), x=0..2*Pi, y = -10..10,  color = green,                                                 thickness = 3, discont = true) );

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