1) This document provides a trigonometry cheat sheet with formulas and identities for tangent, cotangent, sine, cosine, and other trig functions. 2) It includes definitions, properties, and formulas related to right triangles, the unit circle, inverse trig functions, and laws of sines, cosines, and tangents. 3) Examples are provided to demonstrate how to use formulas like half-angle, double-angle, sum and difference, and cofunction identities as well as inverse trig functions and trigonometric equations.

1. Trig Cheat Sheet Formulas and Identities
Tangent and Cotangent Identities Half Angle Formulas
sin θ cos θ
sin 2 θ = (1 − cos ( 2θ ) )
1
Definition of the Trig Functions tan θ = cot θ =
Right triangle definition cos θ sin θ 2
Reciprocal Identities
cos 2 θ = (1 + cos ( 2θ ) )
For this definition we assume that Unit circle definition 1
π For this definition θ is any angle. csc θ =
1
sin θ =
1 2
0 <θ < or 0° < θ < 90° .
2 y sin θ csc θ 1 − cos ( 2θ )
tan θ =
2
sec θ =
1
cos θ =
1 1 + cos ( 2θ )
( x, y ) cos θ sec θ Sum and Difference Formulas
sin (α ± β ) = sin α cos β ± cos α sin β
hypotenuse 1 1 1
y θ cot θ = tan θ =
opposite tan θ cot θ
x
x
Pythagorean Identities cos (α ± β ) = cos α cos β ∓ sin α sin β
θ sin 2 θ + cos 2 θ = 1 tan α ± tan β
tan (α ± β ) =
adjacent tan θ + 1 = sec θ
2 2 1 ∓ tan α tan β
Product to Sum Formulas
opposite hypotenuse y 1 1 + cot 2 θ = csc 2 θ
sin θ = csc θ = sin θ = =y csc θ = 1
sin α sin β = ⎡cos (α − β ) − cos (α + β ) ⎤
2⎣ ⎦
hypotenuse opposite 1 y Even/Odd Formulas
cos θ =
adjacent
sec θ =
hypotenuse x 1 sin ( −θ ) = − sin θ csc ( −θ ) = − cscθ
cos θ = = x sec θ = 1
cos α cos β = ⎡cos (α − β ) + cos (α + β ) ⎤
hypotenuse adjacent 1 x cos ( −θ ) = cos θ sec ( −θ ) = sec θ 2⎣ ⎦
opposite adjacent y x
tan θ = cot θ = tan θ = cot θ = tan ( −θ ) = − tan θ cot ( −θ ) = − cot θ 1
sin α cos β = ⎣sin (α + β ) + sin (α − β ) ⎦
⎡ ⎤
adjacent opposite x y 2
Periodic Formulas 1
cos α sin β = ⎡sin (α + β ) − sin (α − β ) ⎤
Facts and Properties If n is an integer. 2⎣ ⎦
Domain sin (θ + 2π n ) = sin θ csc (θ + 2π n ) = csc θ Sum to Product Formulas
The domain is all the values of θ that Period
cos (θ + 2π n ) = cos θ sec (θ + 2π n ) = sec θ ⎛α + β ⎞ ⎛α −β ⎞
can be plugged into the function. The period of a function is the number, sin α + sin β = 2sin ⎜ ⎟ cos ⎜ ⎟
T, such that f (θ + T ) = f (θ ) . So, if ω tan (θ + π n ) = tan θ cot (θ + π n ) = cot θ ⎝ 2 ⎠ ⎝ 2 ⎠
sin θ , θ can be any angle is a fixed number and θ is any angle we ⎛α + β ⎞ ⎛α − β ⎞
Double Angle Formulas sin α − sin β = 2 cos ⎜ ⎟ sin ⎜ ⎟
cos θ , θ can be any angle ⎝ 2 ⎠ ⎝ 2 ⎠
⎛ 1⎞
have the following periods. sin ( 2θ ) = 2sin θ cos θ
tan θ , θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… ⎛α + β ⎞ ⎛α −β ⎞
⎝ 2⎠ 2π cos ( 2θ ) = cos 2 θ − sin 2 θ cos α + cos β = 2 cos ⎜ ⎟ cos ⎜ ⎟
sin (ωθ ) → T= ⎝ 2 ⎠ ⎝ 2 ⎠
csc θ , θ ≠ n π , n = 0, ± 1, ± 2,… ω = 2 cos 2 θ − 1 ⎛α + β ⎞ ⎛α − β ⎞
⎛ 1⎞ 2π cos α − cos β = −2sin ⎜ ⎟ sin ⎜ ⎟
sec θ , θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… cos (ω θ ) → T = = 1 − 2sin 2 θ ⎝ 2 ⎠ ⎝ 2 ⎠
⎝ 2⎠ ω
π 2 tan θ Cofunction Formulas
cot θ , θ ≠ n π , n = 0, ± 1, ± 2,… tan ( ωθ ) → = tan ( 2θ ) =
T
ω 1 − tan 2 θ ⎛π ⎞ ⎛π ⎞
sin ⎜ − θ ⎟ = cos θ cos ⎜ − θ ⎟ = sin θ
2π ⎝2 ⎠ ⎝2 ⎠
Range csc (ωθ ) → T =
Degrees to Radians Formulas
The range is all possible values to get ω If x is an angle in degrees and t is an ⎛π ⎞ ⎛π ⎞
csc ⎜ − θ ⎟ = sec θ sec ⎜ − θ ⎟ = csc θ
out of the function. 2π angle in radians then ⎝2 ⎠ ⎝2 ⎠
−1 ≤ sin θ ≤ 1 csc θ ≥ 1 and csc θ ≤ −1 sec (ω θ ) → T = π πx
ω =
t
⇒ t= and x =
180t ⎛π ⎞ ⎛π ⎞
tan ⎜ − θ ⎟ = cot θ cot ⎜ − θ ⎟ = tan θ
−1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 π 180 x 180 π ⎝ 2 ⎠ ⎝ 2 ⎠
−∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞ cot (ωθ ) → T =
ω
© 2005 Paul Dawkins © 2005 Paul Dawkins

2. Unit Circle
Inverse Trig Functions
Definition Inverse Properties
cos ( cos −1 ( x ) ) = x
y
( 0,1) y = sin −1 x is equivalent to x = sin y cos−1 ( cos (θ ) ) = θ
π ⎛1 3⎞
⎜ , ⎟
y = cos −1 x is equivalent to x = cos y sin ( sin −1 ( x ) ) = x sin −1 ( sin (θ ) ) = θ
⎛ 1 3⎞ ⎜2 2 ⎟
⎜− , ⎟ ⎝ ⎠ −1
2 y = tan x is equivalent to x = tan y
⎝ 2 2 ⎠ π ⎛ 2 2⎞ tan ( tan −1 ( x ) ) = x tan −1 ( tan (θ ) ) = θ
2π 90° ⎜
⎜ 2 , 2 ⎟
⎟
⎛ 2 2⎞ 3 ⎝ ⎠
⎜− , ⎟
⎝ 2 2 ⎠ 3
120° 60°
π Domain and Range
3π ⎛ 3 1⎞ Function Domain Range Alternate Notation
4 ⎜ , ⎟ sin −1 x = arcsin x
⎛ 3 1⎞
⎜ 2 2⎟
⎝ ⎠ π π
⎜− , ⎟
4
135° 45° π y = sin −1 x −1 ≤ x ≤ 1 − ≤ y≤
⎝ 2 2⎠ 5π 2 2 cos −1 x = arccos x
6 −1
0≤ y ≤π
6 30° y = cos x −1 ≤ x ≤ 1 tan −1 x = arctan x
150° π π
y = tan −1 x −∞ < x < ∞ − < y<
2 2
( −1,0 ) π 180° 0° 0 (1,0 )
Law of Sines, Cosines and Tangents
360° 2π x
c β a
210°
7π 330°
11π
6 225°
⎛ 3 1⎞ 6 ⎛ 3 1⎞ α γ
⎜− ,− ⎟ 5π 315° ⎜ ,− ⎟
⎝ 2 2⎠ ⎝ 2 2⎠
4 240° 300° 7π
⎛ 2⎞ 4π 270° b
5π
2 4
⎜− ,− ⎟ ⎛ 2 2⎞
⎝ 2 2 ⎠ 3 3π ⎜
⎝ 2
,− ⎟
2 ⎠
3 Law of Sines Law of Tangents
⎛ 1 3⎞ 2
⎜ − ,− ⎟
⎛ 1
⎜ ,−
3⎞
⎟ sin α sin β sin γ
= = a − b tan 1 (α − β )
⎝ 2 2 ⎠ ⎝2 2 ⎠ = 2
( 0,−1) a b c a + b tan 1 (α + β )
2
Law of Cosines b − c tan 1 ( β − γ )
= 2
a 2 = b 2 + c 2 − 2bc cos α b + c tan 1 ( β + γ )
2
For any ordered pair on the unit circle ( x, y ) : cos θ = x and sin θ = y b 2 = a 2 + c 2 − 2ac cos β a − c tan 1 (α − γ )
= 2
c = a + b − 2ab cos γ
2 2 2
a + c tan 1 (α + γ )
2
Example
Mollweide’s Formula
⎛ 5π ⎞ 1 ⎛ 5π ⎞ 3
cos ⎜ ⎟ = sin ⎜ ⎟ = − a + b cos 1 (α − β )
⎝ 3 ⎠ 2 ⎝ 3 ⎠ 2 = 2
c sin 1 γ
2
© 2005 Paul Dawkins © 2005 Paul Dawkins