Trigonometric Identities
Definitions and Fundamental Identities:
Circle:
222
xyr
+= sin
y
r
θ= cos
x
r
θ=
sin
tan
cos
y
x
θ
θ==
θ
Pythagorean Relations
22
cossin1
θ+θ=
22
1tansec
+θ=θ
22
1cotcsc
+θ=θ
Reciprocal Relations
1
csc
sin
θ=
θ
1
sin
csc
θ=
θ
11
1
cscsinx
x
−−
=
1
sec
cos
θ=
θ
1
cos
sec
θ=
θ
11
1
seccosx
x
−−
=
1
cot
tan
θ=
θ
1
tan
cot
θ=
θ
11
1
cottanx
x
−−
=
Odd Relations Even Relations
sin()sin
−θ=−θ
tan()tan
−θ=−θ
cos()cos
−θ=θ
csc()csc
−θ=−θ
cot()cot
−θ=−θ
sec()sec
−θ=θ
Sum Relations Difference Relations
sin()sincoscossin
ABABAB
+=+
sin()sincoscossin
ABABAB
−=−
cos()coscossinsin
ABABAB
+=−
cos()coscossinsin
ABABAB
−=+
tantan
tan()
1tantan
AB
AB
AB
+
+=
−
tantan
tan()
1tantan
AB
AB
AB
−
−=
+
Double Angle Relations Half Angle Relations
sin22sincos
θ=θθ
1cos
sin 22
θ−θ
=±
22
2
2
cos2cossin
2cos1
12sin
θ=θ−θ
=θ−
=−θ
1cos
cos 22
θ+θ
=±
2
2tan
tan2
1tan
θ
θ=
−θ
1cos
tan
2sin
θ−θ
=
θ
2
1cos2
sin
2
−θ
θ= sin
1cos
θ
=
+θ
2
1cos2
cos
2
+θ
θ=
1cos
1cos
−θ
=±
+θ
