math made simple :)

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