Trig. Cheat Sheet
Definition of the Six Trigonometric Functions:
Right triangle definitions, where
02
π
θ
<<
Interpreting the unit circle:
( ) ( )
, cos ,sin
5
sin150 sin 0.5
6
xy
θθ
π
=
= =
..
sin , csc .
..
cos , sec
..
..
tan , cot
..
opp hyp
hyp opp
adj hyp
hyp adj
opp adj
adj opp
θθ
θθ
θθ
= =
= =
= =
Defining the opposite and adjacent sides is dependent on the angle.
Circular function definitions, where
θ
is any angle in standard
position and
22
0.r xy= +>
.
sin ,cos ,tan , csc ,sec ,cot
yxyrrx
rr xyxy
θθθ θθθ
= = = = = =
Reciprocal Identities
111
sin cos tan
csc sec cot
111
csc sec cot
sin cos tan
uuu
uuu
uuu
uuu
= = =
= = =
Tangent and Cotangent Identities
sin cos
tan cot
cos sin
uu
uu
uu
= =
Pythagorean Identities
22 22
22
sin cos 1, 1 tan sec ,
1 cot csc
uu uu
uu
+=+=
+=
Cofunction Identities
sin cos , cos sin
22
csc sec , sec csc
22
tan cot , cot tan
22
uu uu
uu uu
uu uu
ππ
ππ
ππ
−= −=
−= −=
−= −=
Sum and Difference Formulas
sin( ) sin cos cos sin
cos( ) cos cos sin sin
tan tan
tan( ) 1 tan tan
uv u v u v
uv u v u v
uv
uv uv
±= ±
±=
±
±=
Double-Angle Formulas
2
22
2
2
2tan
sin 2 2sin cos , tan 2 1 tan
cos2 cos sin
2cos 1
1 2sin
u
u uu u u
u uu
u
u
= = −
= −
= −
= −
Power-Reducing Formulas
22
2
1 cos2 1 cos 2
sin , cos
22
1 cos2
tan 1 cos 2
uu
uu
u
uu
−+
= =
−
=+
Half-angle Formulas
1 cos 1 cos
sin , cos
2 22 2
1 cos sin 1 cos
tan 2 1 cos 1 cos sin
u uu u
u uu u
u uu
−+
=±=±
−−
=±==
++
Arc Length
sr
θ
=
Law of Sines
sin sinA B SinC
abc
= =
Law of Cosines
2 22
2 cosc a b ab C=+−
DeMoivre’s Theorem
(cos sin )
nn
Z r nin
θθ
= +
Parabola:
22
4 , 4x ay y ax= =
Ellipse:
22
22
1
xy
ab
+=
Hyperbola:
22
22
1
xy
ab
−=
Deg. to Rad.
deg. , 45 , 45 0.7854
180 180 4
radians radians
π ππ
×= → ×=≈
Rad. to Deg.
180 180
. deg ., 2.68 . deg., 2.68 153.55rad rad
ππ
×= → ×≈
Negative Angle Identites
sin( ) sin , cos( ) cos , tan( ) tan ,
csc( ) csc , sec( ) sec , cot( ) cot .
u u uu u u
u u uu u u
−=− −= −=−
−=− −= −=−
Tutoring Services, Chemeketa Community College, 4000 Lancaster Dr. NE, Salem, OR 97309.
Chemeketa Community College is an equal opportunity, affirmative action institution. Revised 10/10
θ
