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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
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Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
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the polar co-ordinate of the point (-1, -1)
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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