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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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