Trigonometry

Trigonometry – Key Relationships and Formulas Function Relationships sin θ
cos θ
tan θ
tan θ
1
csc θ
csc θ
1
sec θ
sec θ
1
cot θ
cot θ
1
sin θ
1
cos θ
1
tan θ
sin θ
cot θ
cos θ
cos θ
sin θ
cos
tan
1
sec
cot
1
csc
sin
Double Angle Formulas 2sin cos tan
sin cos 2
cos
1 2 sin 2 cos
1 2 tan
1 tan
tan 2
sin
sin
cos
tan
cot
sec
csc
sin
cos
cos
tan
tan
cot
cot
sec
sec
csc
csc
1 cos
sin 2
Cofunction Formulas (in Quadrant I)
2
2
2
1
1
1
1
1
cos
2
Triple Angle Formulas sin 3
3sin
cos
2
cos 3
4 cos
tan 3
3 tan
tan
1 3 tan
cos
cos
sin
3 cos Law of Sines Law of Cosines sin
2
cos 2
cos 2
cos Euler’s Formula sin
cis cis
csc
sec
2
2
sin
sin cos cos sin sin sin cos cos sin cos cos cos
sin sin cos cos cos
sin sin tan tan
1–tan tan
tan tan
1 tan tan
∙ sin sin
sin
2 ∙ sin
sin
sin
2 ∙ sin
cos
cos
2 ∙ cos
cos
cos
2 ∙ sin
cos
cos
sin
sin
∙ cos
2
∙ cos
2
2
2
∙ cos
2
2
∙ sin
2
2
Mollweide’s Formulas
mathguy.us Polar Multiplication and Division
cis Let: ∙
tan
Sum‐to‐Product Formulas 1
2
1
tan
2
cot
2
tan
DeMoivre’s Formula cis
cos 2
2
cos 2
2
cos 2
cos 2
sin
1
cos
2
1
cos ∙ cos cos
2
1
sin ∙ cos sin
2
1
sin
cos ∙ sin
2
cos
sin
Law of Tangents
cos
1
1
tan
Arc Length 1
cos
cos
Product‐to‐Sum Formulas sin
1 cos
1
2
tan
Power Reducing Formulas
4sin
2
tan 2
Angle Addition Formulas Half Angle Formulas
Pythagorean Identities sin
Opposite Angle Formulas
cis
cis cis
cos
sin
sin
1
2
1
2
cos
1
2
1
2
2 Period Period 2 Period Period 2
Period 2
Period ∙
Amplitude: | | " "
Period: Phase Shift: Vertical Shift: Harmonic Motion
cos
or sin
2
,
0 Trig Functions of Special Angles (Unit Circle) ° Rad 0⁰ 0 1 0 6 30⁰ 1/2 √3/2 √3/3 4 45⁰ √2/2 √2/2 1 3 60⁰ √3/2 1/2 √3 2 90⁰ 1 0 undefined 0 Rectangular/Polar Conversion Rectangular Polar ,
,
cos sin tan
cos
sin
or cos sin tan
tan
Vector Dot Product ∙
∙
∘
∘
Vector Properties 0
0
0 1
2
1
2
1
2
sin sin
sin
1
2
sin 1
2
1
1
2
2
3
3
1
1
1 1
‖
‖
| |‖ ‖ Unit Vector: ‖ ‖
Vector Cross Product
u u
u v
x
v v
x
‖ ‖ ‖ ‖ sin θ x
‖ ‖
‖ ‖ cos ‖ ‖ sin 1
2
‖ ‖∠ ∘
∘
Triangle Area
Angle between Vectors
u v x
cos
iff ‖
∘
∘
sin
‖‖ ‖
‖
‖
‖ ‖‖ ‖
0 ∥ iff x
0