<< Chapter < Page Chapter >> Page >

Reduction formula

Any trigonometric function whose argument is 90 ± θ , 180 ± θ , 270 ± θ and 360 ± θ (hence - θ ) can be written simply in terms of θ . For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of 90 . From this we may expect that sin ( 90 + θ ) = cos θ .

Function values of 180 ± θ

Investigation : reduction formulae for function values of 180 ± θ

  1. Function Values of ( 180 - θ )
    1. In the figure P and P' lie on the circle with radius 2. OP makes an angle θ = 30 with the x -axis. P thus has coordinates ( 3 ; 1 ) . If P' is the reflection of P about the y -axis (or the line x = 0 ), use symmetry to write down the coordinates of P'.
    2. Write down values for sin θ , cos θ and tan θ .
    3. Using the coordinates for P' determine sin ( 180 - θ ) , cos ( 180 - θ ) and tan ( 180 - θ ) .
    1. From your results try and determine a relationship between the function values of ( 180 - θ ) and θ .
  2. Function values of ( 180 + θ )
    1. In the figure P and P' lie on the circle with radius 2. OP makes an angle θ = 30 with the x -axis. P thus has coordinates ( 3 ; 1 ) . P' is the inversion of P through the origin (reflection about both the x - and y -axes) and lies at an angle of 180 + θ with the x -axis. Write down the coordinates of P'.
    2. Using the coordinates for P' determine sin ( 180 + θ ) , cos ( 180 + θ ) and tan ( 180 + θ ) .
    3. From your results try and determine a relationship between the function values of ( 180 + θ ) and θ .

Investigation : reduction formulae for function values of 360 ± θ

  1. Function values of ( 360 - θ )
    1. In the figure P and P' lie on the circle with radius 2. OP makes an angle θ = 30 with the x -axis. P thus has coordinates ( 3 ; 1 ) . P' is the reflection of P about the x -axis or the line y = 0 . Using symmetry, write down the coordinates of P'.
    2. Using the coordinates for P' determine sin ( 360 - θ ) , cos ( 360 - θ ) and tan ( 360 - θ ) .
    3. From your results try and determine a relationship between the function values of ( 360 - θ ) and θ .

It is possible to have an angle which is larger than 360 . The angle completes one revolution to give 360 and then continues to give the required angle. We get the following results:

sin ( 360 + θ ) = sin θ cos ( 360 + θ ) = cos θ tan ( 360 + θ ) = tan θ

Note also, that if k is any integer, then

sin ( k 360 + θ ) = sin θ cos ( k 360 + θ ) = cos θ tan ( k 360 + θ ) = tan θ

Write sin 293 as the function of an acute angle.

  1. sin 293 = sin ( 360 - 67 ) = - sin 67

    where we used the fact that sin ( 360 - θ ) = - sin θ . Check, using your calculator, that these values are in fact equal:

    sin 293 = - 0 , 92 - sin 67 = - 0 , 92

Evaluate without using a calculator:

tan 2 210 - ( 1 + cos 120 ) sin 2 225
  1. tan 2 210 - ( 1 + cos 120 ) sin 2 225 = [ tan ( 180 + 30 ) ] 2 - [ 1 + cos ( 180 - 60 ) ] · [ sin ( 180 + 45 ) ] 2 = ( tan 30 ) 2 - [ 1 + ( - cos 60 ) ] · ( - sin 45 ) 2 = 1 3 2 - 1 - 1 2 · - 1 2 2 = 1 3 - 1 2 1 2 = 1 3 - 1 4 = 1 12

Reduction formulae

  1. Write these equations as a function of θ only:
    1. sin ( 180 - θ )
    2. cos ( 180 - θ )
    3. cos ( 360 - θ )
    4. cos ( 360 + θ )
    5. tan ( 180 - θ )
    6. cos ( 360 + θ )
  2. Write the following trig functions as a function of an acute angle:
    1. sin 163
    2. cos 327
    3. tan 248
    4. cos 213
  3. Determine the following without the use of a calculator:
    1. tan 150 . sin 30 + cos 330
    2. tan 300 . cos 120
    3. ( 1 - cos 30 ) ( 1 - sin 210 )
    4. cos 780 + sin 315 . tan 420
  4. Determine the following by reducing to an acute angle and using special angles. Do not use a calculator:
    1. cos 300
    2. sin 135
    3. cos 150
    4. tan 330
    5. sin 120
    6. tan 2 225
    7. cos 315
    8. sin 2 420
    9. tan 405
    10. cos 1020
    11. tan 2 135
    12. 1 - sin 2 210

Function values of ( - θ )

When the argument of a trigonometric function is ( - θ ) we can add 360 without changing the result. Thus for sine and cosine

sin ( - θ ) = sin ( 360 - θ ) = - sin θ
cos ( - θ ) = cos ( 360 - θ ) = cos θ

Function values of 90 ± θ

Investigation : reduction formulae for function values of 90 ± θ

  1. Function values of ( 90 - θ )
    1. In the figure P and P' lie on the circle with radius 2. OP makes an angle θ = 30 with the x -axis. P thus has coordinates ( 3 ; 1 ) . P' is the reflection of P about the line y = x . Using symmetry, write down the coordinates of P'.
    2. Using the coordinates for P' determine sin ( 90 - θ ) , cos ( 90 - θ ) and tan ( 90 - θ ) .
    3. From your results try and determine a relationship between the function values of ( 90 - θ ) and θ .
  2. Function values of ( 90 + θ )
    1. In the figure P and P' lie on the circle with radius 2. OP makes an angle θ = 30 with the x -axis. P thus has coordinates ( 3 ; 1 ) . P' is the rotation of P through 90 . Using symmetry, write down the coordinates of P'. (Hint: consider P' as the reflection of P about the line y = x followed by a reflection about the y -axis)
    2. Using the coordinates for P' determine sin ( 90 + θ ) , cos ( 90 + θ ) and tan ( 90 + θ ) .
    3. From your results try and determine a relationship between the function values of ( 90 + θ ) and θ .

Complementary angles are positive acute angles that add up to 90 . e.g. 20 and 70 are complementary angles.

Sine and cosine are known as co-functions . Two functions are called co-functions if f ( A ) = g ( B ) whenever A + B = 90 (i.e. A and B are complementary angles). The other trig co-functions are secant and cosecant, and tangent and cotangent.

The function value of an angle is equal to the co-function of its complement (the co-co rule).

Thus for sine and cosine we have

sin ( 90 - θ ) = cos θ cos ( 90 - θ ) = sin θ

Write each of the following in terms of 40 using sin ( 90 - θ ) = cos θ and cos ( 90 - θ ) = sin θ .

  1. cos 50
  2. sin 320
  3. cos 230
    1. cos 50 = cos ( 90 - 40 ) = sin 40
    2. sin 320 = sin ( 360 - 40 ) = - sin 40
    3. cos 230 = cos ( 180 + 50 ) = - cos 50 = - cos ( 90 - 40 ) = - sin 40

Function values of ( θ - 90 )

sin ( θ - 90 ) = - cos θ and cos ( θ - 90 ) = sin θ .

These results may be proved as follows:

sin ( θ - 90 ) = sin [ - ( 90 - θ ) ] = - sin ( 90 - θ ) = - cos θ

and likewise for cos ( θ - 90 ) = sin θ

Summary

The following summary may be made

second quadrant ( 180 - θ ) or ( 90 + θ ) first quadrant ( θ ) or ( 90 - θ )
sin ( 180 - θ ) = + sin θ all trig functions are positive
cos ( 180 - θ ) = - cos θ sin ( 360 + θ ) = sin θ
tan ( 180 - θ ) = - tan θ cos ( 360 + θ ) = cos θ
sin ( 90 + θ ) = + cos θ tan ( 360 + θ ) = tan θ
cos ( 90 + θ ) = - sin θ sin ( 90 - θ ) = sin θ
cos ( 90 - θ ) = cos θ
third quadrant ( 180 + θ ) fourth quadrant ( 360 - θ )
sin ( 180 + θ ) = - sin θ sin ( 360 - θ ) = - sin θ
cos ( 180 + θ ) = - cos θ cos ( 360 - θ ) = + cos θ
tan ( 180 + θ ) = + tan θ tan ( 360 - θ ) = - tan θ
  1. These reduction formulae hold for any angle θ . For convenience, we usually work with θ as if it is acute, i.e. 0 < θ < 90 .
  2. When determining function values of 180 ± θ , 360 ± θ and - θ the functions never change.
  3. When determining function values of 90 ± θ and θ - 90 the functions changes to its co-function (co-co rule).

Function values of ( 270 ± θ )

Angles in the third and fourth quadrants may be written as 270 ± θ with θ an acute angle. Similar rules to the above apply. We get

third quadrant ( 270 - θ ) fourth quadrant ( 270 + θ )
sin ( 270 - θ ) = - cos θ sin ( 270 + θ ) = - cos θ
cos ( 270 - θ ) = - sin θ cos ( 270 + θ ) = + sin θ

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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Other chapter Q/A we can ask
Moahammedashifali Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 11 maths' conversation and receive update notifications?

Ask