Deriving the trigonometric addition formulae using area and cosine rule


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 formulaPlanetmathPlanetmath 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.

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(α±β)=sinαcosβ±sinβcosα (1)


cos(α±β)=cosαcosβsinαsinβ, (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 theoremMathworldPlanetmath 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
area of  XYZ=12xysinθ

and the cosine rule gives


The simple trigonometric identities are given by

sin(180-θ)=sinθ, (3)
cos(180-θ)=-cosθ, (4)
sin2θ+cos2θ=1. (5)

3 Derivation of Addition Trigonometric Formulae

Figure 2: QuadrilateralMathworldPlanetmath ABCD

Fig. (2) shows a quadrilateral ABCD which consists of two right-angled triangles ABC and ACD. In ABC, CA^B=α, AC=s. By simple trigonometryMathworldPlanetmath, we have

AB=scosα,BC=ssinα. (6)

In ACD, CA^D=β, AD=h. We have the following trigonometric equations:

cosβ=sh,CD=hsinβ. (7)

We also note that BC^D=180-α.
We can express the area of quadrilateral ABCD as:

Area ofABC+Area ofACD=Area ofABD+Area ofBCD
                                                                +12(CD)(BC)sin(180-α). (8)

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

12s2sinαcosα+12hssinβ=12hscosαsin(α+β)+12hssinβsin2α. (9)

Dividing eq. (9) by 12hs, we obtain


Simplifying and using eq. (5), we have

cosαsin(α+β) = shsinαcosα+sinβ(1-sin2α), (10)
= shsinαcosα+sinβcos2α.

Dividing eq. (10) by cosα and using eq. (7), we finally obtain the sine addition formula given by eq. (1) with the + sign.
We next consider BCD. By using cosine rule and eqs. (4), (6) and (7), we have

BD2 = CD2+BC2-2(CD)(BC)cos(180-α), (11)
= h2sin2β+s2sin2α+2hssinβsinαcosα.

Using cosine rule in ABD and eq. (11), we obtain

cos(α+β) = h2+AB2-BD22h(AB) (12)
= h2(1-sin2β)+s2(cos2α-sin2α)-2hssinβsinαcosα2hscosα.

Using eq. (5) and s2=h2cos2β, eq. (12) reduces to

cos(α+β) = h2cos2β+s2(cos2α-sin2α)-2hssinβsinαcosα2hscosα (13)
= s2(1+cos2α-sin2α)-2hssinβsinαcosα2hscosα
= 2s2cos2α-2hssinβsinαcosα2hscosα
= shcosα-sinαsinβ.

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 β by -β in eqs. (1) and (2).
But we adopt a different approach. We set γ=α+β so that β=γ-α.
Then eqs. (1) and (2) reduce to

sin(γ)=sinαcos(γ-α)+cosαsin(γ-α) (14)


cosγ=cosαcos(γ-α)-sinαsin(γ-α), (15)

respectively. We obtain the trigonometric difference formulae by solving eqs. (14) and (15) simultaneously.
(14)×cosα-(15)×sinα yields


so that

sin(γ-α)=sinγcosα-cosγsinα (16)

using eq. (5).
Similarly, (14)×sinα+(15)×cosα results in

cos(γ-α)=cosγcosα+sinγsinα. (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
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