Trigonometric Equivalences

Trigonometric Equivalences
Common Values
cos −θ
Reciprocal Relationships
=
sec θ =
cot θ =
= sin(θ + π2 )
cos nπ = sin(nπ + π2 )
sin −θ = − sin θ
csc θ
cos
cos θ
= cos θ
Quotient Relationships
1
sin θ
1
cos θ
1
tan θ
√
π
6
π
sin
6
π
tan
6
Periods
Angles vs. Signs
tan θ =
cot θ =
sin θ
cos θ
cos θ
sin θ
=
=
=
3
2
1
2
√1
3
Pythagorean Relationships
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
Subtraction Formulas
Addition Formulas
sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = sin α cos β − cos α sin β
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
tan(α + β)
=
tan α+tan β
1−tan α tan β
=
tan α−tan β
1+tan α tan β
Half-Angle Formulas
Double-Angle Formulas
sin 2α
= 2 sin α cos α
cos 2α
= cos2 α − sin2 α =
= 1 − 2 sin2 α = 2 cos2 α − 1
tan 2α
tan(α − β)
=
2 tan α
1−tan2 α
Products of Sines and Cosines
q
1
θ
sin θ
= ± 1−cos
2
2
q
1
θ
cos θ
= ± 1+cos
2
2
q
1
θ
sin θ
tan θ = ± 1−cos
1+cos θ = 1+cos θ =
2
1−cos θ
sin θ
Sum and Difference of Sines and Cosines
sin α cos β
= 12 [sin(α + β) + sin(α − β)]
sin α + sin β
= 2 sin 21 (α + β) cos 12 (α − β)
cos α sin β
= 21 [sin(α + β) − sin(α − β)]
sin α − sin β
= 2 cos 12 (α + β) sin 12 (α − β)
cos α cos β = 12 [cos(α + β) + cos(α − β)]
cos α + cos β = 2 cos 12 (α + β) cos 12 (α − β)
sin α sin β = 12 [cos(α + β) − cos(α − β)]
sin α + sin β
Law of Sines
= 2 sin 12 (α + β) sin 12 (α − β)
Law of Cosines
a
b
c
=
=
sin A
sin B
sin C
sin A
sin B
sin C
=
=
a
b
c
a2 = b2 + c2 − 2bc cos A
b2 = a2 + c2 − 2ac cos B
c2 = a2 + b2 − 2ab cos C
Author: Martin Blais, 2009. This work is licensed under the Creative Commons “Attribution - Non-Commercial - Share-Alike” license.