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This lesson submitted by amanda.bragg

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*) =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*

**(2)**

**EXERCISE 1: Basic use of a reduction formula**

Write sin293∘ as the function of an acute angle.

**EXERCISE 2: More complicated...**

Evaluate without using a calculator:

tan2210∘−(1+cos120∘)sin2225∘

**(5)**

**Reduction Formulae**

- Write these equations as a function of
*θ*only: - sin(180∘−
*θ*) - cos(180∘−
*θ*) - cos(360∘−
*θ*) - cos(360∘+
*θ*) - tan(180∘−
*θ*) - cos(360∘+
*θ*) - Write the following trig functions as a function of an acute angle:
- sin163∘
- cos327∘
- tan248∘
- cos213∘
- Determine the following without the use of a calculator:
- tan150∘.sin30∘+cos330∘
- tan300∘.cos120∘
- (1−cos30∘)(1−sin210∘)
- cos780∘+sin315∘.tan420∘
- Determine the following by reducing to an acute angle and using special angles. Do not use a calculator:
- cos300∘
- sin135∘
- cos150∘
- tan330∘
- sin120∘
- tan2225∘
- cos315∘
- sin2420∘
- tan405∘
- cos1020∘
- tan2135∘
- 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*)=−sin*θ*

**(7)**

cos(−*θ*)=cos(360∘−*θ*)=cos*θ*

**(8)**

**Function Values of **90∘±*θ*

**Investigation : Reduction Formulae for Function Values of **90∘±*θ*

**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 reflection of P about the line*y*=*x*. Using symmetry, write down the coordinates of P'. - 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_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*)==cos*θ*sin*θ*

**(9)**

**EXERCISE 3: Co-co rule**

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

- cos50∘
- sin320∘
- cos230∘

**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*

**(10)**

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)=+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