Real-Time Rendering, Third Edition, 3rd Edition

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918 B. Trigonometry

Equations B.12 and B.13 show a collection of laws that we call the angle

sum and angle difference relations.

Angle sum relations :

sin(φ + ρ)=sinφ cos ρ +cosφ sin ρ

cos(φ + ρ)=cosφ cos ρ − sin φ sin ρ

tan(φ + ρ)=

tan φ +tanρ

1 − tan φ tan ρ

(B.12)

Angle difference relations :

sin(φ − ρ)=sinφ cos ρ − cos φ sin ρ

cos(φ −ρ)=cosφ cos ρ +sinφ sin ρ

tan(φ − ρ)=

tan φ − tan ρ

1+tanφ tan ρ

(B.13)

Next follow the product relations.

Product relations :

sin φ sin ρ =

1

2

(cos(φ − ρ) −cos(φ + ρ))

cos φ cos ρ =

1

2

(cos(φ − ρ)+cos(φ + ρ))

sin φ cos ρ =

1

2

(sin(φ − ρ)+sin(φ + ρ))

(B.14)

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B.2. Trigonometric Laws and Formulae 919

The formulae in Equations B.15 and B.16 go under the names function

sums and differences and half-angle relations.

Function sums and differences :

sin φ +sinρ =2sin

φ + ρ

2

cos

φ − ρ

2

cos φ +cosρ =2cos

φ + ρ

2

cos

φ − ρ

2

tan φ +tanρ =

sin(φ + ρ)

cos φ cos ρ

sin φ − sin ρ =2cos

φ + ρ

2

sin

φ − ρ

2

cos φ −cos ρ = −2sin

φ + ρ

2

sin

φ − ρ

2

tan φ − tan ρ =

sin(φ − ρ)

cos φ cos ρ

(B.15)

Half-angle relations :

sin

φ

2

= ±

(

1 − cos φ

2

cos

φ

2

= ±

(

1+cosφ

2

tan

φ

2

= ±

(

1 − cos φ

1+cosφ

=

1 − cos φ

sin φ

=

sin φ

1+cosφ

(B.16)

Further Reading and Resources

The first chapter of Graphics Gems [405] provides other geometric relation-

ships that are useful in computer graphics. The 31st edition of the CRC

Standard Mathematical Tables and Formulas [1416] includes the formulae

in this appendix and much more.

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