<< Chapter < Page Chapter >> Page >
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 gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask