Trigonometry Formula Sheet:

Arc Length:
s=θr θ → radians

Area of a Sector:
1 2
A= θ r θ → radians
2

A= ( 360θ ° ) π r θ →degrees
2

Six Trig Functions: (SOH CAH TOA)

opp hyp
sin θ= csc θ=
hyp opp

adj hyp
cos θ= sec θ=
hyp adj

opp adj
tanθ= cot θ=
adj opp

Graphing Trig Functions:

y= Asin ( Bx+ c )+ D

max−min 2π
Amplitude: | A|= Period: P=
2 B

max + min
Vertical Shift: D= Phase Shift:
2
−C
x=
B

Law of Sines:
sin A sin B sin C
= =
a b c

Law of Cosines:
2 2 2
c =a + b +2 ab cos C

Law of Tangents:

a−b tan [ 1/2( A−B) ]

=
a+b tan [ 1/2( A+ B) ]
Reciprocal Quotient Identities: Pythagorean
Identities: Identities:
sin θ
tanθ=
1 cos θ 2
sin θ+cos θ=1
2
cot θ=
tan θ
cos θ 2
1+ tan θ=sec θ
2
cot θ=
1 sinθ
csc θ=
sin θ 2
1+cot θ=csc θ
2

1
sec θ=
cos θ

Even-Odd Identities: Co-function Power Reducing

Identities: Formulas:
sin (−θ )=−sin θ
cos (−θ )=cos θ cos ( 90° −θ )=sinθ 2 1−cos 2θ
sin θ=
tan (−θ )=−tan θ sin ( 90 °−θ )=cos θ 2
csc (−θ )=−csc θ tan ( 90° −θ )=cotθ
sec (−θ )=sec θ cot ( 90 °−θ )=tanθ 2 1+ cos 2θ
cos θ=
cot (−θ )=−cotθ sec ( 90° −θ )=cscθ 2
csc ( 90 °−θ )=secθ
2 1−cos 2 θ
tan θ=
1+ cos 2 θ

Double Angle Half-Angle Formulas: Triple Angle

Formulas: Formulas:

sin 2 θ=2 sin θ cos θ

sin ( θ2 )=± √ 1−cos
2
θ
sin 3 θ=3 sin θ−4 sin θ
3

2 tan θ _________________________
sin 2 θ= 2
1+ tan θ ___ 3
cos 3 θ=4 cos θ−3 cos θ
_______________________
___
cos ( θ2 )=± √ 1+ cos
2
θ

2 2 3
cos 2 θ=cos θ−sin θ 3 tan θ−tan θ
tan3 θ= 2
1−3 tan θ

( θ2 )= 1−cos θ sin θ
2
cos 2 θ=2 cos −1 tan =
2 1+ cos θ
2
cos 2 θ=1−2 sin θ

cos 2 θ=
1−tan θ
2
1+ tan θ
2
tan ( θ2 )=± √ 1−cos θ
1+cos θ

_______________________
___
 2 tan θ
tan2 θ= 2
1−tan θ

Sum and Difference Identities: Polar Equations:

sin ( α ± β )=sin α cos β ±cos α sin β x=r cos θ y =r sin θ

cos ( α ± β ) =cos α cos β ∓ sin α sin β r =√ x + y

2 2

( yx )
tan α ± tan β
tan(α ± β )= θ=tan −1
1 ∓ tan α tan β

Sum-to-Product Formulas: Product-to-Sum Formulas:

sin α +sin β=2 sin ( α +2 β )cos ( α −β

2 )
1
sin α sin β= ¿
2

1
sin α −sin β=2 sin ( α−β
2 ) ( )
cos
α+β
2
cos α cos β= ¿
2

1
cos α +cos β=2 cos
α +β
2 ( ) (
cos
α −β
2 ) sin α cos β= ¿
2

1
cos α sin β= ¿
cos α−cos β=−2 sin
α +β
2
sin( ) (
α −β
2 ) 2

Area of a Triangle:

1
A= ab sin C
2

Heron’s Formula:

Area=√ s (s−a)(s−b)( s−c )

1
s= (a+ b+c )
2
 Common Trigonometric Values:
Degre Radian sin θ cos θ csc θ sec θ tanθ cot θ
es: s:
0° 0 0 1 Undefine 1 0 Undefine
d d
30 ° π 1 √3 2 2 √3 √3 √3
6 2 2 3 3

45 ° π √2 √2 √2 √2 1 1
4 2 2

60 ° π √3 1 2 √3 2 √3 √3
3 2 2 3 3

90 ° π 1 0 1 Undefine Undefine 0
2 d d

180 ° π 0 −1 Undefine −1 0 Undefine

d d
270 ° 3π −1 0 −1 Undefine Undefine 0
2 d d

360 ° 2π 0 1 Undefine 1 0 Undefine

d d

Common Inverse Trigonometric Values:

 −1 −1 −1
x sin (x) cos (x) tan (x)

−√ 3 N/ A N/ A
−60 °=
−π
3
−1 −π 180 °=π −π
−90 °= −45° =
2 4
−√ 3 −60 °=
−π
150 °=
5π
2 3 6

−√ 2 −45° =
−π
135 °=
3π
2 4 4

−√ 3 −30 °=
−π
3 6

−1 −π 2π
−30 °= 120 °=
2 6 3

0 0° π 0°
90 °=
2
1 π π
30 °= 60 °=
2 6 3

√3 30 °=
π
3 6

√2 45 ° =
π
45 ° =
π
2 4 4

√3 60 °=
π
30 °=
π
2 3 6

1 π 0° π
90 °= 45 ° =
2 4
√3 N/ A N/ A
60 °=
π
3