Trigonometry Formulas

Trigonometry Formulas
y
6
r
r
...
sin θ =
(x, y)
θ
...
..
I
y
r
cos θ =
x
r
tan θ =
y
x
-x
sec θ =
1
cos θ
csc θ =
1
sin θ
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
C
...
.........
....... ..
....... ..
....... ....
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.
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.
.
.
.......
...
.......
...
.......
..
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..
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.
..
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...
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...
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..
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..
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Law of Sines:
b
b
c
a
=
=
sin A
sin B
sin C
a
A
c
Law of Cosines: a2 = b2 + c2 − 2bc cos A
B
θ
30◦ (π/6)
sin θ
cos θ
tan θ
1/2
√
3/2
√
3/3
45◦ (π/4)
√
2/2
√
2/2
1
60◦ (π/3)
√
3/2
1/2
√
3
Pythagorean Identities
sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
cos(−θ) = cos θ
tan(−θ) = − tan θ
Symmetry Identities
sin(−θ) = − sin θ
Sum and Difference Identities
sin(α + β) = sin α cos β + cos α sin β
cos(α + β) = cos α cos β − sin α sin β
tan α + tan β
tan(α + β) =
1 − tan α tan β
sin(α − β) = sin α cos β − cos α sin β
cos(α − β) = cos α cos β + sin α sin β
tan α − tan β
tan(α − β) =
1 + tan α tan β
Double-Angle Identities
cos 2θ = cos2 θ − sin2 θ
= 1 − 2 sin2 θ
sin 2θ = 2 sin θ cos θ
2 tan θ
tan 2θ =
1 − tan2 θ
= 2 cos2 θ − 1
Power-Reducing Identities
1 + cos 2θ
cos2 θ =
2
Product-To-Sum Identities
sin2 θ =
1 − cos 2θ
2
1
sin(α − β) + sin(α + β)
2
1
sin α sin β =
cos(α − β) − cos(α + β)
2
1
cos α cos β =
cos(α − β) + cos(α + β)
2
sin α cos β =
Gilles Cazelais. Typeset with LATEX on April 29, 2006.