Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that π 0 < θ < or 0° < θ < 90° . 2
Unit circle definition For this definition θ is any angle. y
x,
hypotenuse
1
y
opposite
θ x
x
θ adjacent sin θ = cosθ = tan θ =
opposite hypotenuse adjacent hypotenuse opposite adjacent
cscθ = secθ = cot θ =
hypotenuse opposite hypotenuse adjacent adjacent opposite
sin θ = cosθ = tan θ =
y
1 x 1 y
= y
cscθ =
= x
secθ =
x
cot θ =
1 y
1 x x y
Facts and Properties Domain The domain is all the values of θ that can be plugged into the function.
sinθ sin θ , cosθ cosθ ,,
cscθ cscθ ,,
θ can be any angle θ can be any angle 1⎞ ⎛ θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… 2⎠ ⎝ θ ≠ n π , n = 0, ± 1, ± 2,…
secθ secθ ,,
θ ≠ ⎜n+
tan θ , θ ,
cot θ , θ ,
⎛ ⎝
1⎞
⎟ π , n = 0, ± 1, ± 2,… 2⎠ θ ≠ n π , n = 0, ± 1, ± 2,…
Range The range is all possible values to get out of the function. csc θ ≥ 1 and csc θ ≤ −1 −1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 tan θ ≤ ∞ cot θ ≤ ∞ −∞ ≤ tanθ −∞ ≤ cotθ
Period The period of a function is the number, T , such that f (θ + T) = f (θ ) . So, if ω if ω
is a fixed number and θ is any angle we have the following periods. sin (ω θ ) →
T =
cos (ω θ ) →
T =
tan ( ω θ ) →
T =
csc (ω θ ) →
T =
sec (ω θ ) →
T =
cot (ω θ ) →
T =
2π ω 2π ω π ω 2π ω 2π ω π ω
© 2005 Paul Dawkins
Formulas and Identities Tangent and Cotangent Identities sin θ cos θ tan θ = cot θ = cos θ sin θ Reciprocal Identities 1 1 csc θ = sin θ = sin θ csc θ 1 1 sec θ = cos θ = cos θ sec θ 1 1 cot θ = tan θ = tan θ cot θ Pythagorean Identities sin 2 θ + cos 2 θ = 1
Half Angle Formulas 1 sin 2 θ = (1 − cos ( 2θ ) ) 2 1 cos 2 θ = (1 + cos ( 2θ ) ) 2 1 − cos ( 2θ ) tan 2 θ = 1 + cos ( 2θ ) Sum and Difference Formulas sin (α ± β ) = sin α cos β ± cos α sin β
cos (α ± β ) = cos α cos β tan (α ± β ) =
∓ sin α sin
β
tan α ± tan β
1 ∓ tan α tan β Product to Sum Formulas 1 + cot 2 θ = csc 2 θ 1 sin α sin β = ⎡⎣cos (α − β ) − cos (α + β ) ⎤⎦ Even/Odd Formulas 2 sin ( −θ ) = − sin θ csc ( −θ ) = − csc θ 1 cos α cos β = ⎡⎣cos (α − β ) + cos (α + β ) ⎤⎦ cos ( −θ ) = cos θ sec ( −θ ) = sec θ 2 1 tan ( −θ ) = − tan θ cot ( −θ ) = − cot θ sin α cos β = ⎡⎣sin (α + β ) + sin (α − β ) ⎤⎦ 2 Periodic Formulas 1 cos α sin β = ⎡⎣sin (α + β ) − sin (α − β ) ⎤⎦ If n is an integer. 2 sin (θ + 2π n ) = sin θ csc (θ + 2π n ) = csc θ Sum to Product Formulas cos (θ + 2π n ) = cos θ sec (θ + 2π n ) = sec θ ⎛α + β ⎞ ⎛ α − β ⎞ sin α + sin β = 2 sin ⎜ ⎟ cos ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ tan (θ + π n ) = tan θ cot (θ + π n ) = cot θ ⎛ α + β ⎞ ⎛ α − β ⎞ Double Angle Formulas sin α − sin β = 2 cos ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ sin ( 2θ ) = 2 sin θ cos θ ⎛α + β ⎞ ⎛ α − β ⎞ 2 2 α β c o s c o s 2 c o s c o s + = ⎜ ⎟ ⎜ ⎟ cos ( 2θ ) = cos θ − sin θ ⎝ 2 ⎠ ⎝ 2 ⎠ = 2 cos 2 θ − 1 ⎛ α + β ⎞ ⎛ α − β ⎞ cos α − cos β = −2 sin ⎜ 2 ⎟ sin ⎜ 2 ⎟ = 1 − 2 sin θ 2 ⎝ ⎠ ⎝ ⎠ Cofunction Formulas 2tan θ tan ( 2θ ) = 1 − tan 2 θ ⎛π ⎞ ⎛ π ⎞ sin ⎜ − θ ⎟ = cos θ cos ⎜ − θ ⎟ = sin θ ⎝2 ⎠ ⎝2 ⎠ Degrees to Radians Formulas If x x is an angle in degrees and t is an ⎛π ⎞ ⎛ π ⎞ csc ⎜ − θ ⎟ = sec θ sec ⎜ − θ ⎟ = csc θ angle in radians then ⎝2 ⎠ ⎝2 ⎠ π t π x 180t ⎛π ⎞ ⎛ π ⎞ and x = = ⇒ t= tan ⎜ − θ ⎟ = cot θ cot ⎜ − θ ⎟ = tan θ π 180 x 180 ⎝2 ⎠ ⎝2 ⎠ tan 2 θ + 1 = sec 2 θ
© 2005 Paul Dawkins
Unit Circle y
2 2π
⎛ 2 2⎞ , ⎜− ⎟ ⎝ 2 2 ⎠
3
⎛ 2 2⎞ , ⎜⎜ ⎟⎟ ⎝ 2 2 ⎠
3
π
60°
4 45°
135°
5π
π 30°
6
( −1,0)
π
90°
120°
3π 4
⎛1 3⎞ ⎜⎜ , ⎟⎟ ⎝2 2 ⎠
π
⎛ 1 3⎞ ⎜− , ⎟ ⎝ 2 2 ⎠
⎛ 3 1⎞ ⎜− , ⎟ ⎝ 2 2⎠
( 0,1 0,1)
6
150°
0°
0
360°
2π
π 180°
7π ⎛ 3 1⎞ ⎜− ,− ⎟ ⎝ 2 2⎠
⎛ 3 1⎞ ⎜⎜ , ⎟⎟ ⎝ 2 2⎠
210°
6
330° 225°
5π
⎛ 2 2⎞ ⎜ − ,− ⎟ 2 ⎠ ⎝ 2
4
240° 4π
270°
3
3π 2
⎛ 1 3⎞ ⎜ − ,− ⎟ ⎝ 2 2 ⎠
315° 7π
300°
5π
4
3
( 0,−1)
1 0)
x
11π 6
⎛ 3 1⎞ ⎜ ,− ⎟ ⎝ 2 2⎠
⎛ 2 2⎞ ,− ⎜ ⎟ 2 ⎠ ⎝ 2
⎛1 3⎞ ⎜ ,− ⎟ ⎝2 2 ⎠
For any ordered pair on the unit circle ( x, y ) : cos θ = x and sin θ = y Example
⎛ 5π ⎝ 3
cos ⎜
⎞ 1 ⎟= 2 ⎠
3 ⎛ 5π ⎞ =− ⎟ 2 ⎝ 3 ⎠
sin ⎜
© 2005 Paul Dawkins
Inverse Trig Functions Definition −1 y= sin x is equivalent to x= sin y x is equivalent to x= cos y
−1
x is equivalent to x= tan y
Domain and Range Function Domain y = sin
cos ( cos
−1
y= cos y= tan
Inverse Properties
−1
x
−1 ≤ x ≤ 1
y = cos 1 x
−1 ≤ x ≤ 1
−
y = tan
−1
−∞ < x < ∞
x
π 2
< y <
( x ) ) = x
sin ( sin −1 ( x ) ) = x tan ( tan −1 ( x ) ) = x
Range π π − ≤ y ≤ 2 2 0 ≤ y ≤ π
−
−1
−1
(cos (θ ) ) = θ sin − (sin (θ ) ) = θ tan − ( tan (θ ) ) = θ cos
1
1
Alternate Notation sin −1 x = arcsin x
cos −1 x = arccos x − tan 1 x = arctan x
π 2
Law of Sines, Cosines and Tangents
c
β
a
γ
b Law of Sines sin α sin β a
=
b
=
sin γ c
Law of Tangents a − b tan 12 (α − β ) a+b
Law of Cosines 2 2 2 a = b + c − 2bc cos α
b−c
b = a + c − 2ac cos β
a−c
c = a + b − 2ab cos γ
a+c
2
2
2
2
2 2
b+c
=
= =
tan 12 (α + β ) tan 12 ( β − γ ) tan 12 ( β + γ ) tan 12 (α − γ ) tan 12 (α + γ )
Mollweide’s Formula a + b cos 12 (α − β ) c
=
sin 12 γ
© 2005 Paul Dawkins