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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 .

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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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