Deriving the trigonometric addition formulae using area and cosine rule

Abstract

The trigonometric addition formulae are very important and useful in Mathematics. They can be derived from geometric proof, relations from analytical geometry, Euler formula or relations from vectorial analysis. In this work, we prove the sine addition formula by considering area. Using cosine rule, we prove the cosine addition formula. Our proof is simpler because it requires the knowledge of area of triangles and cosine rule which are easily understood by students. From the addition formulae, we obtain the difference formulae by solving simultaneous equations.

Keywords
Trigonometric addition and difference formulae, area of triangles, cosine rule

Deriving the trigonometric addition formulae using area and cosine rule

Deriving the trigonometric addition formulae using area and cosine rule

1 Introduction

The sine and cosine addition and difference formulae are given by

$$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \sin \beta \cos \alpha$$ (1)

and

$$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta ,$$ (2)

respectively. The sine and cosine addition and diference formulae can be obtained from geometric proofs , Euler’s formula, analytical geometry and vectorial analysis . Feldman uses the cosine rule and Pythagoras theorem to get the cosine difference formula. In this work, we consider the area of triangles to obtain the sine addition formula since the area of any triangle depends on the sine of the angle. Then, using the cosine rule twice, we obtain the cosine addition formula. Finally, we obtain the trigonometric difference formulae from addition formulae by solving two simultaneous equations.

2 Preliminaries

We use the important formulas and identities:
From Fig. (1), we have

Figure 1: Triangle XYZ

$$\text{area of }\mathrm{△}\text{ XYZ}=\frac{1}{2}xy\sin \theta$$

and the cosine rule gives

$$z^2=x^2+y^2-2xy\cos \theta .$$

The simple trigonometric identities are given by

$\sin ({180}^{\circ }-\theta )=\sin \theta ,$ (3)

$\cos ({180}^{\circ }-\theta )=-\cos \theta ,$ (4)

${\sin }^2\theta +{\cos }^2\theta =1.$ (5)

3 Derivation of Addition Trigonometric Formulae

Figure 2: Quadrilateral ABCD

Fig. (2) shows a quadrilateral ABCD which consists of two right-angled triangles $ABC$ and $ACD$. In $\mathrm{△}ABC$, $C\widehat{A}B=\alpha$, $AC=s$. By simple trigonometry, we have

(6)

In $\mathrm{△}ACD$, $C\widehat{A}D=\beta$, $AD=h$. We have the following trigonometric equations:

(7)

We also note that $B\widehat{C}D={180}^{\circ }-\alpha .$
We can express the area of quadrilateral ABCD as:

$\text{Area of}\mathrm{△}ABC+\text{Area of}\mathrm{△}ACD=\text{Area of}\mathrm{△}ABD+\text{Area of}\mathrm{△}BCD$

${\displaystyle \frac{1}{2}}(AB)(BC)+{\displaystyle \frac{1}{2}}(AC)(CD)={\displaystyle \frac{1}{2}}(AB)h\sin (\alpha +\beta )$

$\mathrm{ }+{\displaystyle \frac{1}{2}}(CD)(BC)\sin ({180}^{\circ }-\alpha ).$ (8)

Substituting eqs. (3), (6) and (7) into eq. (3), we have, after simplifications,

$$\frac{1}{2}{s}^2\sin \alpha \cos \alpha +\frac{1}{2}hs\sin \beta =\frac{1}{2}hs\cos \alpha \sin (\alpha +\beta )+\frac{1}{2}hs\sin \beta {\sin }^2\alpha .$$ (9)

Dividing eq. (9) by ${\displaystyle \frac{1}{2}}hs,$ we obtain

$$\frac{s}{h}\sin \alpha \cos \alpha +\sin \beta =\cos \alpha \sin (\alpha +\beta )+\sin \beta {\sin }^2\alpha .$$

Simplifying and using eq. (5), we have

$\cos \alpha \sin (\alpha +\beta )$ $=$ ${\displaystyle \frac{s}{h}}\sin \alpha \cos \alpha +\sin \beta (1-{\sin }^2\alpha ),$ (10)

$=$ ${\displaystyle \frac{s}{h}}\sin \alpha \cos \alpha +\sin \beta {\cos }^2\alpha .$

Dividing eq. (10) by $\cos \alpha$ and using eq. (7), we finally obtain the sine addition formula given by eq. (1) with the $+$ sign.
We next consider $\mathrm{△}BCD$. By using cosine rule and eqs. (4), (6) and (7), we have

$B{D}^2$ $=$ $C{D}^2+B{C}^2-2(CD)(BC)\cos ({180}^{\circ }-\alpha ),$ (11)

$=$ $h^2{\sin }^2\beta +s^2{\sin }^2\alpha +2hs\sin \beta \sin \alpha \cos \alpha .$

Using cosine rule in $\mathrm{△}ABD$ and eq. (11), we obtain

$\cos (\alpha +\beta )$ $=$ ${\displaystyle \frac{h^2+A{B}^2-B{D}^2}{2h(AB)}}$ (12)

$=$ ${\displaystyle \frac{h^2(1-{\sin }^2\beta )+s^2({\cos }^2\alpha -{\sin }^2\alpha )-2hs\sin \beta \sin \alpha \cos \alpha }{2hs\cos \alpha }}.$

Using eq. (5) and $s^2=h^2{\cos }^2\beta$, eq. (12) reduces to

$\cos (\alpha +\beta )$ $=$ ${\displaystyle \frac{h^2{\cos }^2\beta +s^2({\cos }^2\alpha -{\sin }^2\alpha )-2hs\sin \beta \sin \alpha \cos \alpha }{2hs\cos \alpha }}$ (13)

$=$ ${\displaystyle \frac{s^2(1+{\cos }^2\alpha -{\sin }^2\alpha )-2hs\sin \beta \sin \alpha \cos \alpha }{2hs\cos \alpha }}$

$=$ ${\displaystyle \frac{2s^2{\cos }^2\alpha -2hs\sin \beta \sin \alpha \cos \alpha }{2hs\cos \alpha }}$

$=$ ${\displaystyle \frac{s}{h}}\cos \alpha -\sin \alpha \sin \beta .$

Substituting eq. (7) into eq. (13), we finally obtain cosine addition formula given by eq. (2) with the $+$ sign. We point out the proof of the cosine addition formula is a generalization to that of Feldman who used $s=h=1.$

4 Derivation of Trigonometric Difference Formulae

The difference formulas can be obtained by replacing $\beta$ by $-\beta$ in eqs. (1) and (2).
But we adopt a different approach. We set $\gamma =\alpha +\beta$ so that $\beta =\gamma -\alpha .$
Then eqs. (1) and (2) reduce to

$$\sin (\gamma )=\sin \alpha \cos (\gamma -\alpha )+\cos \alpha \sin (\gamma -\alpha )$$ (14)

and

$$\cos \gamma =\cos \alpha \cos (\gamma -\alpha )-\sin \alpha \sin (\gamma -\alpha ),$$ (15)

respectively. We obtain the trigonometric difference formulae by solving eqs. (14) and (15) simultaneously.
$(\mathit{\text{<a href="\#S4.E14" title="(14) ‣ 4 Derivation of Trigonometric Difference Formulae ‣ Deriving the trigonometric addition formulae using area and cosine rule" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">14</span></a>}})\times \cos \alpha -(\mathit{\text{<a href="\#S4.E15" title="(15) ‣ 4 Derivation of Trigonometric Difference Formulae ‣ Deriving the trigonometric addition formulae using area and cosine rule" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">15</span></a>}})\times \sin \alpha$ yields

$$\sin \gamma \cos \alpha -\cos \gamma \sin \alpha =({\sin }^2\alpha +{\cos }^2\alpha )\sin (\gamma -\alpha )$$

so that

$$\sin (\gamma -\alpha )=\sin \gamma \cos \alpha -\cos \gamma \sin \alpha$$ (16)

using eq. (5).
Similarly, $(\mathit{\text{<a href="\#S4.E14" title="(14) ‣ 4 Derivation of Trigonometric Difference Formulae ‣ Deriving the trigonometric addition formulae using area and cosine rule" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">14</span></a>}})\times \sin \alpha +(\mathit{\text{<a href="\#S4.E15" title="(15) ‣ 4 Derivation of Trigonometric Difference Formulae ‣ Deriving the trigonometric addition formulae using area and cosine rule" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">15</span></a>}})\times \cos \alpha$ results in

$$\cos (\gamma -\alpha )=\cos \gamma \cos \alpha +\sin \gamma \sin \alpha .$$ (17)

Title	Deriving the trigonometric addition formulae using area and cosine rule
Canonical name	DerivingTheTrigonometricAdditionFormulaeUsingAreaAndCosineRule1
Date of creation	2013-03-11 19:55:01
Last modified on	2013-03-11 19:55:01
Owner	dkrbabajee (19083)
Last modified by	(0)
Numerical id	1
Author	dkrbabajee (0)
Entry type	Definition