Graphs of the Other Trigonometric Functions Section 4.6

Objectives Determine the period and phase shift of the graph of a tangent, cotangent, secant, or cosecant function given an equation. Match a graph to its equation. Determine the domain, range, asymptotes, period and phase shift of the graph of a tangent, cotangent, secant, or cosecant function given a graph. Draw the graphs of a tangent, cotangent, secant, or cosecant function. Determine if the sine, cosine, tangent, cotangent, secant, or cosecant functions are even, odd, or neither.

General Function

Graph the function tan(x) on the interval [―2π, 2π] What is the period? continued on next slide π What is the phase shift? none in the basic function What is the domain? The domain is defined everywhere that the cos(x) is not equal to 0.

Graph the function tan(x) on the interval [―2π, 2π] What is the range? (-∞, ∞) Are there any asymptotes? If yes, what are they? There are many (an infinite number) of asymptotes. They occur anywhere that the cos(x) equals 0. Thus the asymptotes are continued on next slide

Graph the function tan(x) on the interval [―2π, 2π] Is the function even, odd or neither? The tangent function is an odd function. Thus tan(-x) = -tan(x). In terms of the graph, if you rotate the graph 180 degrees, you get a new graph that looks exactly like the original. rotated 180 degreesoriginal

Graph the function cot(x) on the interval [―2π, 2π] What is the period? continued on next slide π none in the basic function The domain is defined everywhere that the sin(x) is not equal to 0. What is the phase shift? What is the domain?

Graph the function cot(x) on the interval [―2π, 2π] What is the range? continued on next slide (-∞, ∞) There are many (an infinite number) of asymptotes. They occur anywhere that the sin(x) equals 0. Thus the asymptotes are Are there any asymptotes? If yes, what are they?

Graph the function cot(x) on the interval [―2π, 2π] Is the function even, odd or neither? The cotangent function is an odd function. Thus cot(-x) = -cot(x). In terms of the graph, if you rotate the graph 180 degrees, you get a new graph that looks exactly like the original. rotated 180 degreesoriginal

General Function

Graph the function sec(x) on the interval [―2π, 2π] What is the period? continued on next slide 2π2π What is the phase shift? none in the basic function What is the domain? The domain is defined everywhere that the cos(x) is not equal to 0.

Graph the function sec(x) on the interval [―2π, 2π] continued on next slide What is the range? (-∞, -1]  [1, ∞) Are there any asymptotes? If yes, what are they? There are many (an infinite number) of asymptotes. They occur anywhere that the cos(x) equals 0. Thus the asymptotes are

Graph the function sec(x) on the interval [―2π, 2π] Is the function even, odd or neither? The secant function is an even function. Thus sec(-x) = sec(x). In terms of the graph, if you reflect the graph over the y-axis, you get a new graph that looks exactly like the original. reflected over y-axisoriginal

Graph the function csc(x) on the interval [―2π, 2π] continued on next slide What is the period? 2π2π none in the basic function The domain is defined everywhere that the sin(x) is not equal to 0. What is the phase shift? What is the domain?

Graph the function csc(x) on the interval [―2π, 2π] continued on next slide What is the range? (-∞, ∞) There are many (an infinite number) of asymptotes. They occur anywhere that the sin(x) equals 0. Thus the asymptotes are Are there any asymptotes? If yes, what are they?

Graph the function csc(x) on the interval [―2π, 2π] Is the function even, odd or neither? The cosecant function is an odd function. Thus csc(-x) = -csc(x). In terms of the graph, if you rotate the graph 180 degrees, you get a new graph that looks exactly like the original. rotated 180 degreesoriginal

Graph You should be able to graph this on your calculator. You should note the vertical stretch by a factor of 2 and the change in the period. The new period is found based on the basic tangent graph equation. For our problem B is 1/4

To graph this on your calculator you would need to type in either Y1 = (1/2)(1/tan(x) OR Y1 = (1/2)(cos(x)/sin(x)) You should note the vertical shrink by a factor of ½ cause by the A from the general form of the cotangent equation. Graph

The general form of the cosecant function equation is below The A will affect the range of the function. The 2 in the problem will cause the function to be vertically stretched by a factor of 2 away from the x-axis. This will also stretch the range by a factor of 2 away from the x-axis. This will give a new range of (-∞, -2]  [2, ∞). in order to graph this equation on your calculator, you would need to graph Y1 = 2(1/sin(x)) or just Y1 = 2/sin(x)

Graph You should be able to graph this on your calculator. You should note the phase shift. The new graph will be shifted to the right. How far it is shifted is found based on the basic tangent graph equation.