Trigonometry: Reduction formulae

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This lesson was written by 
Free High School Science Texts Project
Objectives 
Students will learn the reduction formulae.
Students will apply formulae to functions.
Procedure 

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)=cosθ.

 

Function Values of 180∘±θ

Investigation : Reduction Formulae for Function Values of 180∘±θ

Function Values of (180∘−θ)

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'.

Write down values for sinθ, cosθ and tanθ.

Using the coordinates for P' determine sin(180∘−θ), cos(180∘−θ) and tan(180∘−θ).

 

http://cnx.org/content/m38870/latest/MG11C17_023.png

 

(d): From your results try and determine a relationship between the function values of (180∘−θ) and θ.

Function values of (180∘+θ)

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'.

Using the coordinates for P' determine sin(180∘+θ), cos(180∘+θ) and tan(180∘+θ).

From your results try and determine a relationship between the function values of (180∘+θ) and θ.

 

http://cnx.org/content/m38870/latest/MG11C17_024.png

 

Investigation : Reduction Formulae for Function Values of 360∘±θ

Function values of (360∘−θ)

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'.

Using the coordinates for P' determine sin(360∘−θ), cos(360∘−θ) and tan(360∘−θ).

From your results try and determine a relationship between the function values of (360∘−θ) and θ.

 

http://cnx.org/content/m38870/latest/MG11C17_025.png

 

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) =sinθ

cos(360∘+θ) =cosθ

tan(360∘+θ)= tanθ

Note also, that if k is any integer, then

sin(k360∘+θ) =sinθ

cos(k360∘+θ) =cosθ

tan(k360∘+θ) =tanθ

(2)

EXERCISE 1: Basic use of a reduction formula

Write sin293∘ as the function of an acute angle.

[ SHOW SOLUTION ]

EXERCISE 2: More complicated...

Evaluate without using a calculator:

tan2210∘−(1+cos120∘)sin2225∘

(5)

[ SHOW SOLUTION ]

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. sin163∘
    2. cos327∘
    3. tan248∘
    4. cos213∘
  3. Determine the following without the use of a calculator:
    1. tan150∘.sin30∘+cos330∘
    2. tan300∘.cos120∘
    3. (1−cos30∘)(1−sin210∘)
    4. cos780∘+sin315∘.tan420∘
  4. Determine the following by reducing to an acute angle and using special angles. Do not use a calculator:
    1. cos300∘
    2. sin135∘
    3. cos150∘
    4. tan330∘
    5. sin120∘
    6. tan2225∘
    7. cos315∘
    8. sin2420∘
    9. tan405∘
    10. cos1020∘
    11. tan2135∘
    12. 1−sin2210∘

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)=−sinθ

(7)

cos(−θ)=cos(360∘−θ)=cosθ

(8)

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.

 

http://cnx.org/content/m38870/latest/MG11C17_026.png

 

Function values of (90∘+θ)

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)

Using the coordinates for P' determine sin(90∘+θ), cos(90∘+θ) and tan(90∘+θ).

From your results try and determine a relationship between the function values of (90∘+θ) and θ.

 

http://cnx.org/content/m38870/latest/MG11C17_027.png

 

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(90∘−θ)==cosθsinθ

(9)

EXERCISE 3: Co-co rule

Write each of the following in terms of 40∘ using sin(90∘−θ)=cosθ and cos(90∘−θ)=sinθ.

  1. cos50∘
  2. sin320∘
  3. cos230∘

[ SHOW SOLUTION ]

Function Values of (θ−90∘)

sin(θ−90∘)=−cosθ and cos(θ−90∘)=sinθ.

These results may be proved as follows:

sin(θ−90∘)===sin[−(90∘−θ)]−sin(90∘−θ)−cosθ

(10)

and likewise for cos(θ−90∘)=sinθ

Summary

The following summary may be made

 

second quadrant (180∘−θ) or (90∘+θ)            first quadrant (θ) or (90∘−θ)

sin(180∘−θ)=+sinθ      all trig functions are positive

cos(180∘−θ)=−cosθ     sin(360∘+θ)=sinθ

tan(180∘−θ)=−tanθ     cos(360∘+θ)=cosθ

sin(90∘+θ)=+cosθ       tan(360∘+θ)=tanθ

cos(90∘+θ)=−sinθ       sin(90∘−θ)=sinθ

            cos(90∘−θ)=cosθ

third quadrant (180∘+θ)          fourth quadrant (360∘−θ)

sin(180∘+θ)=−sinθ      sin(360∘−θ)=−sinθ

cos(180∘+θ)=−cosθ     cos(360∘−θ)=+cosθ

tan(180∘+θ)=+tanθ     tan(360∘−θ)=−tanθ

 

TIP:

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∘.

When determining function values of 180∘±θ, 360∘±θ and −θ the functions never change.

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)=−cosθ     sin(270∘+θ)=−cosθ

cos(270∘−θ)=−sinθ     cos(270∘+θ)=+sinθ