Trigonometric Identities Reference Sheet Template
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1
2TRIGONOMETRIC IDENTITIES
The six trigonometric functions:
Sum and product formulas:
=
+
+
−
1
sin cos
[sin (
) sin (
)]
opp
y
hyp
r
1
a
b
a
b
a
b
sin θ =
=
θ
=
=
=
csc
2
θ
hyp
r
opp
y
sin
=
+
−
−
1
cos sin
[sin (
)
sin (
)]
a
b
a
b
a
b
2
adj
x
hyp
r
1
=
+
+
−
cos θ =
=
1
θ
=
=
=
cos cos
[cos (
) cos (
)]
a
b
a
b
a
b
sec
2
θ
hyp
r
adj
x
cos
=
−
−
+
1
sin sin
[cos (
) cos (
)]
a
b
a
b
a
b
θ
2
opp
y
sin
adj
x
1
θ
=
=
=
θ
=
=
=
( ) ( )
cot
tan
+
=
+
−
a b
a b
θ
θ
sin
a
sin
b
2
sin
cos
opp
y
tan
adj
x
cos
2
2
( ) ( )
−
=
+
−
a b
a b
sin
a
sin
b
2
cos
sin
Sum or difference of two angles:
2
2
( ) ( )
±
=
±
+
=
+
−
sin (
)
sin cos
cos sin
a b
a b
a
b
a
b
a
b
cos
a
cos
b
2
cos
cos
2
2
m
±
=
( ) ( )
cos(
a b
) cos cos
a
b
sin sin
a
b
−
= −
+
−
a b
a b
cos
a
cos
b
2
sin
sin
±
2
2
tan
a
tan
b
±
=
tan(
a b
)
1 m
tan tan
a
b
2
=
2
+
2
−
2
cos
Law of cosines:
a
b
c
bc
A
θ
2
tan
θ
=
where A is the angle of a scalene triangle opposite
Double angle formulas:
tan
2
−
θ
2
1
tan
side a.
π
θ
=
θ
θ
θ
=
θ
−
2
sin
2
2
sin cos
cos
2
2
cos
1
° =
Radian measure:
1
radians
8.1 p420
θ
=
θ
−
θ
θ
= −
θ
2
2
2
180
cos
2
1 2
sin
cos
2
cos
sin
°
180
radian =
θ
+
θ
=
1
2
2
Pythagorean Identities:
sin
cos
1
π
θ
+ =
θ
θ
+ =
θ
2
2
2
2
tan
1
sec
cot
1
csc
Reduction formulas:
−
θ
= −
θ
− =
θ
θ
Half angle formulas:
sin(
)
sin
cos(
) cos
1
θ
= −
θ π
−
θ
= −
θ π
−
1
θ
=
+
θ
sin( )
sin(
)
cos( )
cos(
)
θ
=
−
θ
2
2
cos
(
1
cos
2
)
sin
(
1
cos
2
)
θ
=
θ π
−
2
2
−
θ
= −
θ
tan( ) tan(
)
tan(
)
tan
θ
−
θ
θ
+
θ
m
1
cos
1
cos
±
=
±
=
±
π
π
= ±
= ±
cos
x
sin(
x
)
sin
x
cos(
x
)
sin
cos
2
2
2
2
2
2
±
θ
=
θ
±
θ
j
Complex Numbers:
θ
−
θ
θ
−
θ
e
cos
j
sin
1
cos
sin
1
cos
= ±
=
=
tan
θ
=
θ
+
−
θ
θ
=
θ
−
−
θ
+
θ
+
θ
θ
j
j
j
j
1
cos
1
(
e
e
)
sin
(
e
e
)
2
1
cos
1
cos
sin
2
j
2
TRIGONOMETRIC VALUES FOR COMMON ANGLES
θ θ
θ θ
θ θ
θ θ
θ θ
θ θ
Degrees
Radians
sin
cos
tan
cot
sec
csc
0°
0
0
1
0
Undefined
1
Undefined
π/6
1/2
2
30°
3
2 /
3
3 /
3
2
3
3 /
π/4
45°
1
1
2
2 /
2
2 /
2
2
π/3
1/2
2
60°
3
2 /
3
3 /
2
3
3 /
3
π/2
90°
1
0
Undefined
0
Undefined
1
-
-
-1/2
-2
120°
2π/3
3
2 /
2
3
3 /
3
3 /
3
135°
3π/4
-
-1
-1
- 2
2
2 /
2
2 /
2
-
- 3
-
1/2
-
2
150°
5π/6
3
2 /
2
3
3 /
3
3 /
π
0
-1
0
Undefined
-1
Undefined
180°
-
-
-1/2
-2
210°
7π/6
3
3 /
3
2 /
3
2
3
3 /
-
- 2
- 2
225°
5π/4
-
1
1
2
2 /
2
2 /
-
240°
4π/3
-
-1/2
3
-2
3
2 /
3
3 /
2
3
3 /
-1
0
Undefined
0
Undefined
-1
270°
3π/2
- 3
-
-
- 3
300°
5π/3
1/2
2
3
2 /
2
3
3 /
-1
-1
- 2
315°
7π/4
-
2
2 /
2
2 /
2
-
- 3
330°
11π/6
-1/2
-2
3
2 /
3
3 /
2
3
3 /
0
1
0
Undefined
1
Undefined
360°
2π
Tom Penick
2/20/2000
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