# 5.2 Reduction formulae

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## Reduction formula

Any trigonometric function whose argument is ${90}^{\circ }±\theta$ , ${180}^{\circ }±\theta$ , ${270}^{\circ }±\theta$ and ${360}^{\circ }±\theta$ (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 $sin\left({90}^{\circ }+\theta \right)=cos\theta$ .

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

1. Function Values of $\left({180}^{\circ }-\theta \right)$
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 $\left(\sqrt{3};1\right)$ . 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 $sin\left({180}^{\circ }-\theta \right)$ , $cos\left({180}^{\circ }-\theta \right)$ and $tan\left({180}^{\circ }-\theta \right)$ .
1. From your results try and determine a relationship between the function values of $\left({180}^{\circ }-\theta \right)$ and $\theta$ .
2. Function values of $\left({180}^{\circ }+\theta \right)$
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 $\left(\sqrt{3};1\right)$ . 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 $sin\left({180}^{\circ }+\theta \right)$ , $cos\left({180}^{\circ }+\theta \right)$ and $tan\left({180}^{\circ }+\theta \right)$ .
3. From your results try and determine a relationship between the function values of $\left({180}^{\circ }+\theta \right)$ and $\theta$ .

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

1. Function values of $\left({360}^{\circ }-\theta \right)$
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 $\left(\sqrt{3};1\right)$ . 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 $sin\left({360}^{\circ }-\theta \right)$ , $cos\left({360}^{\circ }-\theta \right)$ and $tan\left({360}^{\circ }-\theta \right)$ .
3. From your results try and determine a relationship between the function values of $\left({360}^{\circ }-\theta \right)$ 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:

$\begin{array}{ccc}\hfill sin\left({360}^{\circ }+\theta \right)& =& sin\theta \hfill \\ \hfill cos\left({360}^{\circ }+\theta \right)& =& cos\theta \hfill \\ \hfill tan\left({360}^{\circ }+\theta \right)& =& tan\theta \hfill \end{array}$

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

$\begin{array}{ccc}\hfill sin\left(k{360}^{\circ }+\theta \right)& =& sin\theta \hfill \\ \hfill cos\left(k{360}^{\circ }+\theta \right)& =& cos\theta \hfill \\ \hfill tan\left(k{360}^{\circ }+\theta \right)& =& tan\theta \hfill \end{array}$

Write $sin{293}^{\circ }$ as the function of an acute angle.

1. $\begin{array}{ccc}\hfill sin{293}^{\circ }& =& sin\left({360}^{\circ }-{67}^{\circ }\right)\hfill \\ & =& -sin{67}^{\circ }\hfill \end{array}$

where we used the fact that $sin\left({360}^{\circ }-\theta \right)=-sin\theta$ . Check, using your calculator, that these values are in fact equal:

$\begin{array}{ccc}\hfill sin{293}^{\circ }& =& -0,92\cdots \hfill \\ \hfill -sin{67}^{\circ }& =& -0,92\cdots \hfill \end{array}$

Evaluate without using a calculator:

${tan}^{2}{210}^{\circ }-\left(1+cos{120}^{\circ }\right){sin}^{2}{225}^{\circ }$
1. $\begin{array}{ccc}& & {tan}^{2}{210}^{\circ }-\left(1+cos{120}^{\circ }\right){sin}^{2}{225}^{\circ }\hfill \\ & =& {\left[tan\left({180}^{\circ }+{30}^{\circ }\right)\right]}^{2}-\left[1+cos\left({180}^{\circ }-{60}^{\circ }\right)\right]·{\left[sin\left({180}^{\circ }+{45}^{\circ }\right)\right]}^{2}\hfill \\ & =& {\left(tan{30}^{\circ }\right)}^{2}-\left[1+\left(-cos{60}^{\circ }\right)\right]·{\left(-sin{45}^{\circ }\right)}^{2}\hfill \\ & =& {\left(\frac{1}{\sqrt{3}}\right)}^{2}-\left(1,-,\frac{1}{2}\right)·{\left(-,\frac{1}{\sqrt{2}}\right)}^{2}\hfill \\ & =& \frac{1}{3}-\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\hfill \\ & =& \frac{1}{3}-\frac{1}{4}=\frac{1}{12}\hfill \end{array}$

## Reduction formulae

1. Write these equations as a function of $\theta$ only:
1. $sin\left({180}^{\circ }-\theta \right)$
2. $cos\left({180}^{\circ }-\theta \right)$
3. $cos\left({360}^{\circ }-\theta \right)$
4. $cos\left({360}^{\circ }+\theta \right)$
5. $tan\left({180}^{\circ }-\theta \right)$
6. $cos\left({360}^{\circ }+\theta \right)$
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. $tan\phantom{\rule{4pt}{0ex}}{150}^{\circ }.sin\phantom{\rule{4pt}{0ex}}{30}^{\circ }\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}cos\phantom{\rule{4pt}{0ex}}{330}^{\circ }$
2. $tan\phantom{\rule{4pt}{0ex}}{300}^{\circ }.cos\phantom{\rule{4pt}{0ex}}{120}^{\circ }$
3. $\left(1-cos\phantom{\rule{4pt}{0ex}}{30}^{\circ }\right)\left(1-sin\phantom{\rule{4pt}{0ex}}{210}^{\circ }\right)$
4. $cos\phantom{\rule{4pt}{0ex}}{780}^{\circ }\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}sin\phantom{\rule{4pt}{0ex}}{315}^{\circ }.tan\phantom{\rule{4pt}{0ex}}{420}^{\circ }$
4. Determine the following by reducing to an acute angle and using special angles. Do not use a calculator:
1. $cos\phantom{\rule{4pt}{0ex}}{300}^{\circ }$
2. $sin\phantom{\rule{4pt}{0ex}}{135}^{\circ }$
3. $cos\phantom{\rule{4pt}{0ex}}{150}^{\circ }$
4. $tan\phantom{\rule{4pt}{0ex}}{330}^{\circ }$
5. $sin\phantom{\rule{4pt}{0ex}}{120}^{\circ }$
6. ${tan}^{2}{225}^{\circ }$
7. $cos\phantom{\rule{4pt}{0ex}}{315}^{\circ }$
8. ${sin}^{2}{420}^{\circ }$
9. $tan\phantom{\rule{4pt}{0ex}}{405}^{\circ }$
10. $cos\phantom{\rule{4pt}{0ex}}{1020}^{\circ }$
11. ${tan}^{2}{135}^{\circ }$
12. $1-{sin}^{2}{210}^{\circ }$

## Function values of $\left(-\theta \right)$

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

$sin\left(-\theta \right)=sin\left({360}^{\circ }-\theta \right)=-sin\theta$
$cos\left(-\theta \right)=cos\left({360}^{\circ }-\theta \right)=cos\theta$

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

1. Function values of $\left({90}^{\circ }-\theta \right)$
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 $\left(\sqrt{3};1\right)$ . 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 $sin\left({90}^{\circ }-\theta \right)$ , $cos\left({90}^{\circ }-\theta \right)$ and $tan\left({90}^{\circ }-\theta \right)$ .
3. From your results try and determine a relationship between the function values of $\left({90}^{\circ }-\theta \right)$ and $\theta$ .
2. Function values of $\left({90}^{\circ }+\theta \right)$
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 $\left(\sqrt{3};1\right)$ . 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 $sin\left({90}^{\circ }+\theta \right)$ , $cos\left({90}^{\circ }+\theta \right)$ and $tan\left({90}^{\circ }+\theta \right)$ .
3. From your results try and determine a relationship between the function values of $\left({90}^{\circ }+\theta \right)$ 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

$\begin{array}{ccc}\hfill sin\left({90}^{\circ }-\theta \right)& =& cos\theta \hfill \\ \hfill cos\left({90}^{\circ }-\theta \right)& =& sin\theta \hfill \end{array}$

Write each of the following in terms of ${40}^{\circ }$ using $sin\left({90}^{\circ }-\theta \right)=cos\theta$ and $cos\left({90}^{\circ }-\theta \right)=sin\theta$ .

1. $cos{50}^{\circ }$
2. $sin{320}^{\circ }$
3. $cos{230}^{\circ }$
1. $cos{50}^{\circ }=cos\left({90}^{\circ }-{40}^{\circ }\right)=sin{40}^{\circ }$
2. $sin{320}^{\circ }=sin\left({360}^{\circ }-{40}^{\circ }\right)=-sin{40}^{\circ }$
3. $cos{230}^{\circ }=cos\left({180}^{\circ }+{50}^{\circ }\right)=-cos{50}^{\circ }=-cos\left({90}^{\circ }-{40}^{\circ }\right)=-sin{40}^{\circ }$

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

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

These results may be proved as follows:

$\begin{array}{ccc}\hfill sin\left(\theta -{90}^{\circ }\right)& =& sin\left[-\left({90}^{\circ }-\theta \right)\right]\hfill \\ & =& -sin\left({90}^{\circ }-\theta \right)\hfill \\ & =& -cos\theta \hfill \end{array}$

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

## Summary

The following summary may be made

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

## Function values of $\left({270}^{\circ }±\theta \right)$

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

 third quadrant $\left({270}^{\circ }-\theta \right)$ fourth quadrant $\left({270}^{\circ }+\theta \right)$ $sin\left({270}^{\circ }-\theta \right)=-cos\theta$ $sin\left({270}^{\circ }+\theta \right)=-cos\theta$ $cos\left({270}^{\circ }-\theta \right)=-sin\theta$ $cos\left({270}^{\circ }+\theta \right)=+sin\theta$

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