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Formula sheet for trigonometric identities, Cheat Sheet of Trigonometry

Formula sheet include fundamental identities, Pythagorean, reciprocal, odd, even, sum, difference, half angle and double angle relations.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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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
θ+θ=
22
1tansec
+θ
22
1cotcsc
+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
csc()csc
θ=−θ
cot()cot
θ=−θ
sec()sec
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
2
2tan
tan2
1tan
θ
θ=
−θ
1cos
tan
2sin
θ−θ
=
θ
2
1cos2
sin
2
−θ
θ= sin
1cos
θ
=
2
1cos2
cos
2
θ=
1cos
1cos
−θ

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Trigonometric Identities

Definitions and Fundamental Identities:

Circle:

2 2 2 x + y = r sin

y

r

θ = cos

x

r

θ =

sin tan cos

y

x

θ θ = = θ

Pythagorean Relations

2 2 cos θ + sin θ = 1

2 2 1 + tan θ = sec θ

2 2 1 + cot θ = csc θ

Reciprocal Relations

csc sin

θ = θ

sin csc

θ = θ

csc x sin x

− − ^ 

sec cos

θ = θ

cos sec

θ = θ

sec x cos x

− − ^ 

cot tan

θ = θ

tan cot

θ = θ

cot x tan x

− − ^ 

Odd Relations Even Relations

sin( −θ) = − sinθ tan( −θ ) = − tanθ cos( −θ) = cosθ

csc( −θ) = − cscθ cot( −θ) = − cotθ sec( −θ ) = secθ

Sum Relations Difference Relations

sin( A + B ) = sin A cos B + cos A sin B sin( AB ) = sin A cos B −cos A sin B

cos( A + B ) = cos A cos B − sin A sin B cos( AB ) = cos A cos B + sin A sin B

tan tan tan( ) 1 tan tan

A B

A B

A B

tan tan tan( ) 1 tan tan

A B

A B

A B

Double Angle Relations Half Angle Relations

s i n 2 θ = 2sin θ cosθ

1 cos sin 2 2

θ − θ = ±

2 2

2

2

cos2 cos sin

2 c o s 1

1 2sin

θ = θ − θ

= θ −

= − θ

1 cos cos 2 2

θ + θ = ±

2

2tan t a n 2

1 tan

θ θ =

− θ

1 cos tan 2 sin

θ − θ

θ

2 1 c o s 2 sin 2

− θ θ =

sin

1 cos

θ

  • θ

2 1 c o s 2 cos 2

  • θ θ =

1 cos

1 cos

− θ = ±

  • θ