Trigonometry Formulas

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Math Formulas: Trigonometry Identities
Right-Triangle Definitions
1.
sin α =
Opposite
Hypotenuse
2.
cos α =
Adjacent
Hypotenuse
3.
tan α =
Opposite
Adjacent
4.
csc α =
1
Hypotenuse
=
sin α
Opposite
5.
sec α =
1
Hypotenuse
=
cos α
Adjacent
6.
cot α =
1
Adjacent
=
tan α
Opposite
Reduction Formulas
7.
sin(−x) = − sin(x)
8.
cos(−x) = cos(x)
9.
sin
10.
cos
11.
sin
12.
cos
π
2
π
2
π
2
π
2
− x = cos(x)
− x = sin(x)
+ x = cos(x)
+ x = − sin(x)
13.
sin(π − x) = sin(x)
14.
cos(π − x) = − cos(x)
15.
sin(π + x) = − sin(x)
16.
cos(π + x) = − cos(x)
Basic Identities
17.
sin2 x + cos2 x = 1
18.
tan2 x + 1 =
1
cos2 x
19.
cot2 x + 1 =
1
sin2 x
Sum and Difference Formulas
1
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20.
sin(α + β) = sin α · cos β + sin β · cos α
21.
sin(α − β) = sin α · cos β − sin β · cos α
22.
cos(α + β) = cos α · cos β − sin α · cos β
23.
cos(α − β) = cos α · cos β + sin α · cos β
24.
tan(α + β) =
tan α + tan β
1 − tan α · tan β
25.
tan(α − β) =
tan α − tan β
1 + tan α · tan β
Double Angle and Half Angle Formulas
26.
sin(2 α) = 2 · sin α · cos α
27.
cos(2 α) = cos2 α − sin2 α
2 tan α
1 − tan2 α
r
1 − cos α
=±
2
r
1 + cos α
=±
2
28.
tan(2 α) =
29.
sin
α
2
30.
cos
α
2
31.
tan
32.
α
1 − cos α
sin α
=
=
2
sin α
1 − cos α
r
α
1 + cos α
tan = ±
2
1 − cos α
Other Useful Trig Formulas
Law of sines
33.
sin α
sin β
sin γ
=
=
α
β
γ
Law of cosines
a2 = b2 + c2 − 2 · b · c · cos α
34.
b2 = a2 + c2 − 2 · a · c · cos β
c2 = a2 + b2 − 2 · a · b · cos γ
Area of triangle
35.
A=
1
a b sin γ
2
2