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Let us look back at our values for tan θ

θ 0 30 45 60 90 180
tan θ 0 1 3 1 3 0

Now that we have graphs for sin θ and cos θ , there is an easy way to visualise the tangent graph. Let us look back at our definitions of sin θ and cos θ for a right-angled triangle.

sin θ cos θ = opposite hypotenuse adjacent hypotenuse = opposite adjacent = tan θ

This is the first of an important set of equations called trigonometric identities . An identity is an equation, which holds true for any value which is put into it. In this case we have shown that

tan θ = sin θ cos θ

for any value of θ .

So we know that for values of θ for which sin θ = 0 , we must also have tan θ = 0 . Also, if cos θ = 0 our value of tan θ is undefined as we cannot divide by 0. The graph is shown in [link] . The dashed vertical lines are at the values of θ where tan θ is not defined.

The graph of tan θ .

Functions of the form y = a tan ( x ) + q

In the figure below is an example of a function of the form y = a tan ( x ) + q .

The graph of 2 tan θ + 1 .

Functions of the form y = a tan ( θ ) + q :

  1. On the same set of axes, plot the following graphs:
    1. a ( θ ) = tan θ - 2
    2. b ( θ ) = tan θ - 1
    3. c ( θ ) = tan θ
    4. d ( θ ) = tan θ + 1
    5. e ( θ ) = tan θ + 2
    Use your results to deduce the effect of q .
  2. On the same set of axes, plot the following graphs:
    1. f ( θ ) = - 2 · tan θ
    2. g ( θ ) = - 1 · tan θ
    3. h ( θ ) = 0 · tan θ
    4. j ( θ ) = 1 · tan θ
    5. k ( θ ) = 2 · tan θ
    Use your results to deduce the effect of a .

You should have found that the value of a affects the steepness of each of the branches. The larger the absolute magnitude of a , the quicker the branches approach their asymptotes, the values where they are not defined. Negative a values switch the direction of the branches. You should have also found that the value of q affects the vertical shift as for sin θ and cos θ . These different properties are summarised in [link] .

Table summarising general shapes and positions of graphs of functions of the form y = a tan ( x ) + q .
a > 0 a < 0
q > 0
q < 0

Domain and range

The domain of f ( θ ) = a tan ( θ ) + q is all the values of θ such that cos θ is not equal to 0. We have already seen that when cos θ = 0 , tan θ = sin θ cos θ is undefined, as we have division by zero. We know that cos θ = 0 for all θ = 90 + 180 n , where n is an integer. So the domain of f ( θ ) = a tan ( θ ) + q is all values of θ , except the values θ = 90 + 180 n .

The range of f ( θ ) = a tan θ + q is { f ( θ ) : f ( θ ) ( - , ) } .

Intercepts

The y -intercept, y i n t , of f ( θ ) = a tan ( x ) + q is again simply the value of f ( θ ) at θ = 0 .

y i n t = f ( 0 ) = a tan ( 0 ) + q = a ( 0 ) + q = q

Asymptotes

As θ approaches 90 , tan θ approaches infinity. But as θ is undefined at 90 , θ can only approach 90 , but never equal it. Thus the tan θ curve gets closer and closer to the line θ = 90 , without ever touching it. Thus the line θ = 90 is an asymptote of tan θ . tan θ also has asymptotes at θ = 90 + 180 n , where n is an integer.

Graphs of trigonometric functions

  1. Using your knowldge of the effects of a and q , sketch each of the following graphs, without using a table of values, for θ [ 0 ; 360 ]
    1. y = 2 sin θ
    2. y = - 4 cos θ
    3. y = - 2 cos θ + 1
    4. y = sin θ - 3
    5. y = tan θ - 2
    6. y = 2 cos θ - 1
  2. Give the equations of each of the following graphs:

The following presentation summarises what you have learnt in this chapter. Ignore the last slide.

End of chapter exercises

  1. Calculate the unknown lengths
  2. In the triangle P Q R , P R = 20  cm, Q R = 22  cm and P R ^ Q = 30 . The perpendicular line from P to Q R intersects Q R at X . Calculate
    1. the length X R ,
    2. the length P X , and
    3. the angle Q P ^ X
  3. A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder?
  4. A ladder of length 25 m is resting against a wall, the ladder makes an angle 37 to the wall. Find the distance between the wall and the base of the ladder?
  5. In the following triangle find the angle A B ^ C
  6. In the following triangle find the length of side C D
  7. A ( 5 ; 0 ) and B ( 11 ; 4 ) . Find the angle between the line through A and B and the x-axis.
  8. C ( 0 ; - 13 ) and D ( - 12 ; 14 ) . Find the angle between the line through C and D and the y-axis.
  9. A 5 m ladder is placed 2 m from the wall. What is the angle the ladder makes with the wall?
  10. Given the points: E(5;0), F(6;2) and G(8;-2), find angle F E ^ G .
  11. An isosceles triangle has sides 9 cm , 9 cm and 2 cm . Find the size of the smallest angle of the triangle.
  12. A right-angled triangle has hypotenuse 13 mm . Find the length of the other two sides if one of the angles of the triangle is 50 .
  13. One of the angles of a rhombus ( rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter 20 cm is 30 .
    1. Find the sides of the rhombus.
    2. Find the length of both diagonals.
  14. Captain Hook was sailing towards a lighthouse with a height of 10 m .
    1. If the top of the lighthouse is 30 m away, what is the angle of elevation of the boat to the nearest integer?
    2. If the boat moves another 7 m towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer?
  15. (Tricky) A triangle with angles 40 , 40 and 100 has a perimeter of 20 cm . Find the length of each side of the triangle.

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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