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In this section, you will:
  • Analyze the graph of  y=tan x.
  • Graph variations of  y=tan x.
  • Analyze the graphs of  y=sec x  and  y=csc x.
  • Graph variations of  y=sec x  and  y=csc x.
  • Analyze the graph of  y=cot x.
  • Graph variations of  y=cot x.

We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.

Analyzing the graph of y = tan x

We will begin with the graph of the tangent    function, plotting points as we did for the sine and cosine functions. Recall that

tan x = sin x cos x

The period    of the tangent function is π because the graph repeats itself on intervals of k π where k is a constant. If we graph the tangent function on π 2 to π 2 , we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.

We can determine whether tangent is an odd or even function by using the definition of tangent.

tan ( x ) = sin ( x ) cos ( x ) Definition of tangent .               = sin x cos x Sine is an odd function, cosine is even .               = sin x cos x The quotient of an odd and an even function is odd .               = tan x Definition of tangent .

Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in [link] .

x π 2 π 3 π 4 π 6 0 π 6 π 4 π 3 π 2
tan ( x ) undefined 3 –1 3 3 0 3 3 1 3 undefined

These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when π 3 < x < π 2 , we can use a table to look for a trend. Because π 3 1.05 and π 2 1.57 , we will evaluate x at radian measures 1.05 < x < 1.57 as shown in [link] .

x 1.3 1.5 1.55 1.56
tan     x 3.6 14.1 48.1 92.6

As x approaches π 2 , the outputs of the function get larger and larger. Because y = tan x is an odd function, we see the corresponding table of negative values in [link] .

x −1.3 −1.5 −1.55 −1.56
tan x −3.6 −14.1 −48.1 −92.6

We can see that, as x approaches π 2 , the outputs get smaller and smaller. Remember that there are some values of x for which cos x = 0. For example, cos ( π 2 ) = 0 and cos ( 3 π 2 ) = 0. At these values, the tangent function is undefined, so the graph of y = tan x has discontinuities at x = π 2  and  3 π 2 . At these values, the graph of the tangent has vertical asymptotes. [link] represents the graph of y = tan x . The tangent is positive from 0 to π 2 and from π to 3 π 2 , corresponding to quadrants I and III of the unit circle.

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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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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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?
Joseph Reply
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
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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Magreth
progressive wave
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Mujahid
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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