Math 1330 Section 5.3 Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions
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1 Math 1330 Section 5.3 Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions In this section, you will learn to graph the rest of the trigonometric functions. We can use some information from the graphs of g ( x) and h( x) to help gather information about the graph of f ( x) = tan( x). sin( x) From the identity tan( x ) =, we can conclude that the graph of the tangent function cos( x) will have vertical asymptotes whenever cos( x) = 0 and the graph of the tangent function will have an x intercept whenever sin( x ) = 0. π 3π 5π cos( x) = 0 when x = ±, ±, ±,, so this is where the graph of f (x) will have vertical asymptotes and sin( x) = 0 when x = 0, ± π, ± 2π, so this gives the location of the zeros of the function. π π We know that tan = 1 and tan = 1, so this gives two additional points on the 4 4 π π graph of the function f ( x) = tan( x) on the interval,. The tangent function is 2 2 periodic with period π, so these values will repeat themselves at multiples of π. So π π tan + k π = 1 and tan + kπ = 1 where k is an integer. 4 4 Using all of this information, we can generate a graph:
2 Often you will need to graph the function over just one period. In this case, you ll use the π π interval,. Here s the graph of f ( x) = tan( x) over this interval, with pertinent 2 2 points marked. Likewise, we can use some information from the graphs of g ( x) and h( x) to help gather information about the graph of f ( x) = cot( x). cos( x) From the identity cot( x ) =, we can conclude that the graph of the cotangent sin( x) function will have vertical asymptotes whenever sin( x) = 0 and the graph of the cotangent function will have an x intercept whenever cos( x ) = 0. sin( x) = 0 when x = 0, ± π, ± 2π,, so this is where the graph of f (x) will have vertical π 3π 5π asymptotes and cos( x) = 0 when x = ±, ±, ±, so this gives the location of the zeros of the function. π 3π We know that cot = 1 and cot = 1, so this gives two additional points on the 4 4 graph of the function f ( x) = cot( x) on the interval ( 0, π ). The cotangent function is periodic with period π, so these values will repeat themselves at multiples of π. So π 3π cot + k π = 1 and cot + kπ = 1 where k is an integer. 4 4
3 Using all of this information, we can generate a graph: Often you will need to graph the function over just one period. In this case, you ll use the 0, π. Here s the graph of f ( x) = cot( x) over this interval. interval ( ) You can take the graph of either of these basic functions and draw the graph of a more complicated function by making adjustments to the key elements of the basic function. The key elements will be the location(s) of the asymptote(s), x intercepts, and the π - π 3π translations of the points at, 1 and either, 1 or, To graph f ( x) = A tan( Bx C) + D, you will start by locating the asymptotes. To do so, π π set Bx C = and Bx C =. Next, find the x coordinate of the point halfway 2 2 between the asymptotes. Evaluate the function at this value to find the location of the
4 translated x intercept. Next, find the x coordinates of the points halfway between the asymptotes and and the translated zero. Evaluate the function at these values to find two more points on the graph of the function. To graph g ( x) = Acot( Bx C) + D, you will start by locating the asymptotes. To do so, set Bx C = 0 and Bx C = π. Next, find the x coordinate of the point halfway between the asymptotes. Evaluate the function at this value to find the location of the translated x intercept. Next, find the x coordinates of the points halfway between the asymptotes and and the translated zero. Evaluate the function at these values to find two more points on the graph of the function. π For both functions, the period will be. You will find vertical shifts and phase shifts as B you did for translations of sine and cosine functions. You ll also be able to take advantage of what you know about the graph of f ( x) 1 to help you graph g ( x) = csc( x). Using the identity csc( x ) =, you can conclude sin( x) that the graph of g will have a vertical asymptote whenever sin( x ) = 0. This means that the graph of g will have vertical asymptotes at x = 0, ± π, ± 2π,. The easiest way to draw a graph of g ( x) = csc( x) is to draw the graph of f ( x), sketch asymptotes at each of the zeros of f ( x), then sketch in the cosecant graph. Here s the graph of f ( x) on the interval 5 π 5, π 2 2. Next, we ll include the asymptotes for the cosecant graph at each point where sin( x ) = 0.
5 Now we ll include the graph of the cosecant function. Typically, you ll just graph over one period (, 2π ) 0. You ll also be able to take advantage of what you know about the graph of 1 f ( x) to help you graph g ( x) = sec( x). Using the identity sec( x ) =, you cos( x) can conclude that the graph of g will have a vertical asymptote whenever cos( x ) = 0.
6 π 3π This means that the graph of g will have vertical asymptotes at x = ±, ±,. The 2 2 easiest way to draw a graph of g ( x) = sec( x) is to draw the graph of f ( x), sketch asymptotes at each of the zeros of f ( x), then sketch in the secant graph. 5π 5π Here s the graph of f ( x) on the interval,. 2 2 Next, we ll include the asymptotes for the secant graph.
7 Now we ll include the graph of the secant function. Typically, you ll just graph over one period (, 2π ) 0. To graph a more complicated secant or cosecant function, it s easiest to draw the graph of the underlying cosine or sine function. You can then draw in asymptotes at the points that are the translations of the zeros of the underlying functions and use the framework to sketch the secant or cosecant function. Example 1: Sketch x f ( x) = 5 tan 2
8 Example 2: Sketch f ( x) = 3cot(2x) π Example 3: Sketch f ( x) = 2cot 3x f π 2 Example 4: Sketch ( x) = tan( 2 x) + 3
9 Example 5: Sketch x f ( x) = 3sec 2 Example 6: Sketch f ( x) = 3csc(2x) + 1
10 πx π Example 7: Sketch f ( x) = 4sec 3 2 2
x,,, (All real numbers except where there are
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