5.2 Reduction formulae

Reduction formula

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

Function values of ${180}^{\circ }\pm \theta$

Investigation : reduction formulae for function values of ${180}^{\circ }\pm \theta$

1. Function Values of $({180}^{\circ }-\theta )$
   1. In the figure P and P' lie on the circle with radius 2. OP makes an angle $\theta ={30}^{\circ }$ with the $x$ -axis. P thus has coordinates $(\sqrt{3};1)$ . 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 $sin\theta$ , $cos\theta$ and $tan\theta$ .
   3. Using the coordinates for P' determine $sin({180}^{\circ }-\theta )$ , $cos({180}^{\circ }-\theta )$ and $tan({180}^{\circ }-\theta )$ .

   

   1. (d) From your results try and determine a relationship between the function values of $({180}^{\circ }-\theta )$ and $\theta$ .
2. Function values of $({180}^{\circ }+\theta )$
   1. In the figure P and P' lie on the circle with radius 2. OP makes an angle $\theta ={30}^{\circ }$ with the $x$ -axis. P thus has coordinates $(\sqrt{3};1)$ . P' is the inversion of P through the origin (reflection about both the $x$ - and $y$ -axes) and lies at an angle of ${180}^{\circ }+\theta$ with the $x$ -axis. Write down the coordinates of P'.
   2. Using the coordinates for P' determine $sin({180}^{\circ }+\theta )$ , $cos({180}^{\circ }+\theta )$ and $tan({180}^{\circ }+\theta )$ .
   3. From your results try and determine a relationship between the function values of $({180}^{\circ }+\theta )$ and $\theta$ .

   

Investigation : reduction formulae for function values of ${360}^{\circ }\pm \theta$

1. Function values of $({360}^{\circ }-\theta )$
   1. In the figure P and P' lie on the circle with radius 2. OP makes an angle $\theta ={30}^{\circ }$ with the $x$ -axis. P thus has coordinates $(\sqrt{3};1)$ . 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 $sin({360}^{\circ }-\theta )$ , $cos({360}^{\circ }-\theta )$ and $tan({360}^{\circ }-\theta )$ .
   3. From your results try and determine a relationship between the function values of $({360}^{\circ }-\theta )$ and $\theta$ .

   

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

Note also, that if $k$ is any integer, then

Reduction formulae

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

Function values of $(-\theta )$

When the argument of a trigonometric function is $(-\theta )$ we can add ${360}^{\circ }$ without changing the result. Thus for sine and cosine

Function values of ${90}^{\circ }\pm \theta$

Investigation : reduction formulae for function values of ${90}^{\circ }\pm \theta$

1. Function values of $({90}^{\circ }-\theta )$
   1. In the figure P and P' lie on the circle with radius 2. OP makes an angle $\theta ={30}^{\circ }$ with the $x$ -axis. P thus has coordinates $(\sqrt{3};1)$ . 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 $sin({90}^{\circ }-\theta )$ , $cos({90}^{\circ }-\theta )$ and $tan({90}^{\circ }-\theta )$ .
   3. From your results try and determine a relationship between the function values of $({90}^{\circ }-\theta )$ and $\theta$ .

   

2. Function values of $({90}^{\circ }+\theta )$
   1. In the figure P and P' lie on the circle with radius 2. OP makes an angle $\theta ={30}^{\circ }$ with the $x$ -axis. P thus has coordinates $(\sqrt{3};1)$ . P' is the rotation of P through ${90}^{\circ }$ . 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 $sin({90}^{\circ }+\theta )$ , $cos({90}^{\circ }+\theta )$ and $tan({90}^{\circ }+\theta )$ .
   3. From your results try and determine a relationship between the function values of $({90}^{\circ }+\theta )$ and $\theta$ .

   

Complementary angles are positive acute angles that add up to ${90}^{\circ }$ . e.g. ${20}^{\circ }$ and ${70}^{\circ }$ are complementary angles.

Sine and cosine are known as co-functions . Two functions are called co-functions if $f\left(A\right)=g\left(B\right)$ whenever $A+B={90}^{\circ }$ (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

Function values of $(\theta -{90}^{\circ })$

$sin(\theta -{90}^{\circ })=-cos\theta$ and $cos(\theta -{90}^{\circ })=sin\theta$ .

These results may be proved as follows:

and likewise for $cos(\theta -{90}^{\circ })=sin\theta$

Summary

The following summary may be made

Function values of $({270}^{\circ }\pm \theta )$

Angles in the third and fourth quadrants may be written as ${270}^{\circ }\pm \theta$ with $\theta$ an acute angle. Similar rules to the above apply. We get