| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
Trigonometric Function | Reciprocal Trigonometric Function |
---|---|
y=sinx | y=sinx1 |
y=cosx | y=cosx1 |
y=tanx | y=tanx1 |
This lesson will explore the graphs of these reciprocal trigonometric functions.
Here are a few recommended readings before getting started with this lesson.
A massive communication tower is anchored to the ground with wires.
These wires are attached to the tower at a height of 5 meters above the ground. The following function models the length of a wire.Let P be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cotangent function, denoted by cot, is defined as the ratio of the x-coordinate to the y-coordinate of P.
cotθ=sinθcosθ
Since division by 0 is not defined, the graph of the parent cotangent function y=cotx has vertical asymptotes where sinx=0. This means that the graph has vertical asymptotes at every multiple of π. The graph of y=cotx can be drawn by making a table of values.
Consider now the general form of a cotangent function.
y=acotbx
Here, a and b are non-zero real numbers and x is measured in radians. The properties of the cotangent function are stated below.
Properties of y=acotbx | |
---|---|
Amplitude | No amplitude |
Number of Cycles in [0,2π] | 2∣b∣ |
Period | ∣b∣π |
Domain | All real numbers except multiples of ∣b∣π |
Range | All real numbers |
Asymptotes occur at the end of each cycle. Therefore, the given function has asymptotes at θ=0 and θ=2π.
Divide the period into fourths and locate the three equidistant points between the asymptotes. The period for this function goes from 0 to 2π, so a table of values will be made for θ=2π, θ=π, and θ=23π.
θ | 3cot(21θ) | y |
---|---|---|
2π | 3cot(21⋅2π) | 3 |
π | 3cot(21⋅π) | 0 |
23π | 3cot(21⋅23π) | -3 |
The points found in the table are (2π,3), (π,0), and (23π,-3).
Finally, the points can be connected with a smooth curve to draw the graph for one cycle.
Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Here, another cycle is graphed.
Find the period and cycle, then graph the asymptotes and plot some points. Finally, sketch the curve.
Next, divide the period into fourths and locate three equidistant points between the asymptotes. Since a period goes from 0 to 2π, a table of values will be made for x=8π, x=4π, and x=83π.
x | 21cot2x | y |
---|---|---|
8π | 21cot2(8π) | 21 |
4π | 21cot2(4π) | 0 |
83π | 21cot2(83π) | -21 |
The points found in the table are (8π,21), (4π,0), and (83π,-21). These three points can be plotted on the plane.
The points are then connected with a smooth curve to graph one period of the function.
Finally, the cycle can be replicated as many times as desired. In this case, the graph will be drawn for values of x between 0 and 2π.
Let P be the point of intersection of the terminal side of an angle in standard position and the unit circle. The secant function, denoted as sec, is defined as the reciprocal of the x-coordinate of P.
secθ=cosθ1
Since division by 0 is not defined, the graph of the parent secant function y=secx has vertical asymptotes where cosx=0. This means that the graph has vertical asymptotes at odd multiples of x=2π. The graph of y=secx can be drawn by making a table of values.
Consider the general form of a secant function.
y=asecbx
Here, a and b are non-zero real numbers and x is measured in radians. The properties of the secant function are be stated in the table below.
Properties of y=asecbx | |
---|---|
Amplitude | No amplitude |
Number of Cycles in [0,2π] | ∣b∣ |
Period | ∣b∣2π |
Domain | All real numbers except odd multiples of 2∣b∣π |
Range | (-∞,-∣a∣]∪[∣a∣,∞) |
Here, the period is 4π and the asymptotes occur every 2π radians. Therefore, the asymptotes are located not only at the beginning and at the end of each period, but also in the middle of each period. Divide the interval between two asymptotes in fourths by the following pattern.
Notice that the middle point in the pattern, the maximum or minimum point, was already plotted in the previous step. The remaining four points can be found for the interval that goes from -π to 3π by making a table of values.
θ | 2sec21θ | y |
---|---|---|
-2π | 2sec(21(-2π)) | 22≈2.83 |
2π | 2sec(21⋅2π) | 22≈2.83 |
23π | 2sec(21⋅23π) | -22≈-2.83 |
25π | 2sec(21⋅25π) | -22≈-2.83 |
Finally, the points can be connected with a smooth curve to draw the graph for one cycle.
Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Here, one more cycle will be graphed.
Graph the related cosine function and recall that the asymptotes of the secant function occur at the zeros of the cosine function.
x | 2secx | y |
---|---|---|
-45π | 2sec(-45π) | -22≈-2.83 |
-43π | 2sec(-43π) | -22≈-2.83 |
-4π | 2sec(-4π) | 22≈2.83 |
4π | 2sec4π | 22≈2.83 |
43π | 2sec43π | -22≈-2.83 |
45π | 2sec45π | -22≈-2.83 |
47π | 2sec47π | 22≈2.83 |
49π | 2sec49π | 22≈2.83 |
The points found in the table can now be plotted. Finally, each set of points can be connected with smooth curves.
The graph of the curve of the bowl has been drawn.
Let P be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cosecant function, denoted as csc, is defined as the reciprocal of the y-coordinate of P.
cscθ=sinθ1
Since division by 0 is not defined, the graph of the parent cosecant function y=cscx has vertical asymptotes where sinx=0. This means that the graph has vertical asymptotes at multiples of π. The graph of y=cscx can be drawn by making a table of values.
Consider the general form of a cosecant function.
y=acscbx
Here, a and b are non-zero real numbers and x is measured in radians. The properties of the cosecant function are stated below.
Properties of y=acscbx | |
---|---|
Amplitude | No amplitude |
Number of Cycles in [0,2π] | ∣b∣ |
Period | ∣b∣2π |
Domain | All real numbers except multiples of ∣b∣π |
Range | (-∞,-∣a∣]∪[∣a∣,∞) |
Here, the period is 4π and the asymptotes occur every 2π radians. Therefore, the asymptotes are located not only at the beginning and at the end of each period, but also in the middle of each period. Divide the interval between two asymptotes in fourths by the following pattern.
Notice that the middle point, which is the maximum or minimum point, was already plotted in the previous step. Four more points can be found for the interval that goes from -2π to 2π by making a table of values.
θ | 2csc21θ | y |
---|---|---|
-23π | 2csc(21(-23π)) | -22≈-2.83 |
-2π | 2csc(21(-2π)) | -22≈-2.83 |
2π | 2csc(21⋅2π) | 22≈2.83 |
23π | 2csc(21⋅23π) | 22≈2.83 |
Finally, each set of points can be connected with a smooth curve to draw the graph for one cycle.
Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Another cycle is graphed below.
Graph the related sine function and recall that the asymptotes of the cosecant function occur at the zeros of the sine function.
x | 21cscx | y |
---|---|---|
-35π | 21csc(-35π) | 33≈0.58 |
-34π | 21csc(-34π) | 33≈0.58 |
-32π | 21csc(-32π) | -33≈-0.58 |
-3π | 21csc(-3π< |