Trig cheat sheet

Trig Cheat Sheet

1. © 2005 Paul Dawkins
Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition
For this definition we assume that
0
2
π
θ<< or 0 90θ°< < °.


opposite
sin
hypotenuse
θ=
hypotenuse
csc
opposite
θ=
adjacent
cos
hypotenuse
θ=
hypotenuse
sec
adjacent
θ=
opposite
tan
adjacent
θ=
adjacent
cot
opposite
θ=


Unit circle definition
For this definition
θ is any angle.

sin
1
y
yθ==
1
csc
y
θ=
cos
1
x
xθ==
1
sec
x
θ=
tan
y
xθ=
cot
x
yθ=
Facts and Properties
Domain
The domain is all the values of
θ that
can be plugged into the function.

sinθ , θ can be any angle
cosθ, θ can be any angle
tanθ,
1
,0,1,2,
2
nn
θπ
⎛⎞
≠+ =±±
⎜⎟
⎝⎠
…
csc
θ, ,0,1,2,nnθπ≠=±± …
secθ,
1
,0,1,2,
2
nn
θπ
⎛⎞
≠+ =±±
⎜⎟
⎝⎠
…
cot
θ, ,0,1,2,nnθπ≠=±± …

Range
The range is all possible values to get
out of the function.
1sin 1
θ−≤ ≤ csc 1 andcsc 1θθ≥≤−
1 cos 1θ−≤ ≤ sec 1 andsec 1θθ≥≤−
tanθ−∞ ≤ ≤ ∞ cot θ−∞ ≤ ≤ ∞

Period
The period of a function is the number,
T, such that ( )()fTfθθ+= . So, if ω
is a fixed number and θ is any angle we
have the following periods.

()sinωθ→
2
T
π
ω
=
()cosωθ→
2
T
π
ω
=
()tanωθ→ T
π
ω
=
()cscωθ→
2
T
π
ω
=
()secωθ→
2
T
π
ω
=
()cotωθ→ T
π
ω
=
θ
adjacent
opposite
hypotenuse
x
y
(),xy
θ
x
y
1

2. © 2005 Paul Dawkins
Formulas and Identities
Tangent and Cotangent Identities
sin cos
tan cot
cos sin
θθ
θθ
θθ==
Reciprocal Identities
11
csc sin
sin csc
11
sec cos
cos sec
11
cot tan
tan cot
θθ
θθ
θθ
θθ
θθ
θθ
==
==
==

Pythagorean Identities
22
22
22
sin cos 1
tan 1 sec
1 cot cscθθ
θθ
θθ
+=
+=
+=

Even/Odd Formulas
() ()
() ()
() ()sin sin csc csc
cos cos sec sec
tan tan cot cotθθ θθ
θθ θθ
θθ θθ−=− −=−
−= −=
−=− −=−

Periodic Formulas
If n is an integer.
() ()
() ()
() ()sin 2 sin csc 2 csc
cos 2 cos sec 2 sec
tan tan cot cot
nn
nn
nnθπ θ θπ θ
θπ θ θπ θ
θπ θ θπ θ+= +=
+= +=
+= +=
Double Angle Formulas
()
()
()
22
2
2
2
sin 2 2sin cos
cos 2 cos sin
2cos 1
12sin
2tan
tan 2
1tanθθθ
θθθ
θ
θ
θ
θ
θ
=
=−
=−
=−
=
−

Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then
180
and
180 180
tx t
tx
x
π π
π
=⇒= =
Half Angle Formulas
()()
()()
()
()
2
2
21
sin 1 cos 2
2
1
cos 1 cos 2
2
1 cos 2
tan
1 cos 2
θθ
θθ
θ
θ
θ=−
=+
−
=
+

Sum and Difference Formulas
( )
()
()
sin sin cos cos sin
cos cos cos sin sin
tan tan
tan
1tantanαβ α β α β
αβ α β α β
αβ
αβ
αβ±= ±
±=
±
±=
∓
∓

Product to Sum Formulas
()()
()()
()()
()()
1
sin sin cos cos
2
1
cos cos cos cos
2
1
sin cos sin sin
2
1
cos sin sin sin
2
αβ αβ αβ
αβ αβ αβ
αβ αβ αβ
αβ αβ αβ=−−+⎡⎤
⎣⎦
=−++⎡⎤
⎣⎦
=++−⎡⎤
⎣⎦
=+−−⎡⎤
⎣⎦
Sum to Product Formulas
sin sin 2sin cos
22
sin sin 2cos sin
22
cos cos 2cos cos
22
cos cos 2sin sin
22
αβ αβ
αβ
αβ αβ
αβ
αβ αβ
αβ
αβ αβ
αβ+−⎛⎞⎛⎞
+=
⎜⎟⎜⎟
⎝⎠⎝⎠
+−⎛⎞⎛⎞
−=
⎜⎟⎜⎟
⎝⎠⎝⎠
+−⎛⎞⎛⎞
+=
⎜⎟⎜⎟
⎝⎠⎝⎠
+−⎛⎞⎛⎞
−=−
⎜⎟⎜⎟
⎝⎠⎝⎠
Cofunction Formulas
sin cos cos sin
22
csc sec sec csc
22
tan cot cot tan
22
ππ
θθ θθ
ππ
θθ θθ
ππ
θθ θθ⎛⎞ ⎛⎞
−= −=
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⎛⎞ ⎛⎞
−= −=
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⎛⎞ ⎛⎞
−= −=
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠

3. © 2005 Paul Dawkins



Unit Circle



For any ordered pair on the unit circle
(),xy : cosxθ= and sinyθ=

Example
51 5 3
cos sin
32 3 2ππ⎛⎞ ⎛⎞
==−
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠



3
π

4
π

6
π

22
,
22
⎛⎞
⎜⎟
⎜⎟
⎝⎠

31
,
22
⎛⎞
⎜⎟
⎜⎟
⎝⎠

13
,
22
⎛⎞
⎜⎟
⎜⎟
⎝⎠

60°
45°
30°
2
3
π

3
4
π

5
6
π

7
6
π
5
4
π
4
3
π

11
6
π

7
4
π

5
3
π

2
π

π
3
2
π

0
2π
13
,
22
⎛⎞
−⎜⎟
⎝⎠

22
,
22
⎛⎞
−⎜⎟
⎝⎠

31
,
22
⎛⎞
−⎜⎟
⎝⎠

31
,
22
⎛⎞
−−⎜⎟
⎝⎠

22
,
22
⎛⎞
−−⎜⎟
⎝ ⎠
13
,
22
⎛⎞
−−⎜⎟
⎝⎠
31
,
22
⎛⎞
−⎜⎟
⎝⎠

22
,
22
⎛⎞
−⎜⎟
⎝⎠

13
,
22
⎛⎞
−⎜⎟
⎝⎠

()0,1
()0, 1−
()1,0−
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360°
0°
x
()1,0
y

4. © 2005 Paul Dawkins




Inverse Trig Functions
Definition
1
1
1
sin is equivalent to sin
cos is equivalent to cos
tan is equivalent to tan
yx xy
yx xy
yx xy
−
−
−
==
==
==


Domain and Range
Function Domain Range
1
sinyx
−
= 11x−≤ ≤
22
y
ππ
−≤≤
1
cosyx
−
= 11x−≤ ≤ 0yπ≤≤
1
tanyx
−
= x−∞ < < ∞
22
y
ππ
−<<

Inverse Properties
()( ) ()()
()() ()()
()() ()()
11
11
11
cos cos cos cos
sin sin sin sin
tan tan tan tan
xx
xx
xx θθ
θθ
θθ
−−
−−
−−
==
==
==


Alternate Notation
1
1
1
sin arcsin
cos arccos
tan arctan
xx
xx
x x
−
−
−
=
=
=

Law of Sines, Cosines and Tangents


Law of Sines
sin sin sin
abc
αβγ
==
Law of Cosines
222
222
222
2cos
2cos
2cos
abc bc
bac ac
cab abα
β
γ=+−
=+−
=+−

Mollweide’s Formula
()
1
2
1
2
cos
sin
ab
cαβ
γ−+
=
Law of Tangents
( )
()
()
()
()
()
1
2
1
2
1
2
1
2
1
2
1
2
tan
tan
tan
tan
tan
tan
ab
ab
bc
bc
ac
acαβ
αβ
βγ
βγ
αγ
αγ−−
=
++
−−
=
++
−−
=
++

c a
b
α
β
γ