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Trigonometry Formulas and Properties Cheat Sheet, Cheat Sheet of Calculus

courtesy of Crafton Hills College (CHC)

Typology: Cheat Sheet

2020/2021

Uploaded on 04/26/2021

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Updated: October 2019
Reciprocal Identities:
sin 𝜃𝜃=1
csc 𝜃𝜃
csc 𝜃𝜃=1
sin 𝜃𝜃
cos 𝜃𝜃=1
sec 𝜃𝜃
sec 𝜃𝜃=1
cos 𝜃𝜃
tan 𝜃𝜃=1
cot 𝜃𝜃
cot 𝜃𝜃=1
tan 𝜃𝜃
Trigonometry Formulas and Properties
Tangent and Cotangent Identities:
tan 𝜃𝜃=sin 𝜃𝜃
cos 𝜃𝜃
cot =cos 𝜃𝜃
sin 𝜃𝜃
Pythagorean Identities:
sin2𝜃𝜃+cos2𝜃𝜃= 1
tan2𝜃𝜃+ 1 = sec2𝜃𝜃
1 + cot2𝜃𝜃=csc2𝜃𝜃
sin(−𝜃𝜃)=sin 𝜃𝜃
cos(−𝜃𝜃)=cos 𝜃𝜃
tan(𝜃𝜃)=tan 𝜃𝜃
csc(−𝜃𝜃)=csc 𝜃𝜃
sec(−𝜃𝜃)=sec 𝜃𝜃
cot(𝜃𝜃)=cot 𝜃𝜃
Cofunction Formulas:
sin 𝜋𝜋
2𝜃𝜃=cos 𝜃𝜃
csc 𝜋𝜋
2𝜃𝜃=sec 𝜃𝜃
tan 𝜋𝜋
2𝜃𝜃=cot 𝜃𝜃
cos 𝜋𝜋
2𝜃𝜃=sin 𝜃𝜃
sec 𝜋𝜋
2𝜃𝜃=csc 𝜃𝜃
cot 𝜋𝜋
2𝜃𝜃=tan 𝜃𝜃
sin 𝛼𝛼sin 𝛽𝛽=1
2[cos(𝛼𝛼𝛽𝛽)cos(𝛼𝛼+𝛽𝛽)]
cos 𝛼𝛼cos 𝛽𝛽=1
2[cos(𝛼𝛼𝛽𝛽)+cos(𝛼𝛼+𝛽𝛽)]
sin 𝛼𝛼cos 𝛽𝛽=1
2[sin(𝛼𝛼+𝛽𝛽)+sin(𝛼𝛼𝛽𝛽)]
cos 𝛼𝛼sin 𝛽𝛽=1
2[sin(𝛼𝛼+𝛽𝛽)sin(𝛼𝛼𝛽𝛽)]
Sum to Product Formulas:
sin 𝛼𝛼+sin 𝛽𝛽= 2 sin 𝛼𝛼+𝛽𝛽
2cos 𝛼𝛼𝛽𝛽
2
sin 𝛼𝛼sin 𝛽𝛽= 2 cos 𝛼𝛼+𝛽𝛽
2sin 𝛼𝛼𝛽𝛽
2
cos 𝛼𝛼+cos 𝛽𝛽= 2 cos 𝛼𝛼+𝛽𝛽
2cos 𝛼𝛼𝛽𝛽
2
cos 𝛼𝛼cos 𝛽𝛽=2sin 𝛼𝛼+𝛽𝛽
2sin 𝛼𝛼𝛽𝛽
2
:
sin(𝛼𝛼±𝛽𝛽) = sin𝛼𝛼cos 𝛽𝛽±sin 𝛽𝛽cos 𝛼𝛼
cos(𝛼𝛼±𝛽𝛽) = cos𝛼𝛼cos 𝛽𝛽sin 𝛼𝛼sin 𝛽𝛽
tan(𝛼𝛼±𝛽𝛽) = tan 𝛼𝛼±tan 𝛽𝛽
1tan 𝛼𝛼tan 𝛽𝛽
Periodic Formulas
:
sin(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=sin 𝜃𝜃
csc(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=csc 𝜃𝜃
cos(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=cos 𝜃𝜃
sec(𝜃𝜃+ 2𝜋𝜋𝜋𝜋)=sec 𝜃𝜃
tan(𝜃𝜃+𝜋𝜋𝜋𝜋)=tan 𝜃𝜃
cot(𝜃𝜃+𝜋𝜋𝜋𝜋)=cot 𝜃𝜃
Half-Angle Formulas:
sin 𝜃𝜃
2= ±1cos 𝜃𝜃
2
cos 𝜃𝜃
2= ±1 + cos 𝜃𝜃
2
tan 𝜃𝜃
2= ±1cos 𝜃𝜃
1 + cos 𝜃𝜃
Double Angle Formulas:
sin(2𝜃𝜃)= 2 sin 𝜃𝜃cos 𝜃𝜃
cos(2𝜃𝜃)=cos2𝜃𝜃sin2𝜃𝜃
= 2 cos
2
𝜃𝜃1
= 1 2sin2𝜃𝜃
tan 2𝜃𝜃=2tan 𝜃𝜃
1tan2𝜃𝜃
pf2

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Updated: October 2019

Reciprocal Identities:

sin 𝜃𝜃 =

csc 𝜃𝜃

csc 𝜃𝜃 =

sin 𝜃𝜃

cos 𝜃𝜃 =

sec 𝜃𝜃

sec 𝜃𝜃 =

cos 𝜃𝜃

tan 𝜃𝜃 =

cot 𝜃𝜃

cot 𝜃𝜃 =

tan 𝜃𝜃

Trigonometry Formulas and Properties

Tangent and Cotangent Identities:

tan 𝜃𝜃 =

sin 𝜃𝜃

cos 𝜃𝜃

cot =

cos 𝜃𝜃

sin 𝜃𝜃

Pythagorean Identities:

sin

2

𝜃𝜃 + cos

2

tan

2

𝜃𝜃 + 1 = sec

2

1 + cot

2

𝜃𝜃 = csc

2

Even/Odd Formulas:

sin

= − sin 𝜃𝜃 cos

= cos 𝜃𝜃 tan

= − tan 𝜃𝜃

csc(−𝜃𝜃) = − csc 𝜃𝜃 sec(−𝜃𝜃) = sec 𝜃𝜃 cot(−𝜃𝜃) = − cot 𝜃𝜃

Cofunction Formulas:

sin �

− 𝜃𝜃� = cos 𝜃𝜃 csc �

− 𝜃𝜃� = sec 𝜃𝜃 tan �

− 𝜃𝜃� = cot 𝜃𝜃

cos �

− 𝜃𝜃� = sin 𝜃𝜃 sec �

− 𝜃𝜃� = csc 𝜃𝜃 cot �

− 𝜃𝜃� = tan 𝜃𝜃

Product to Sum Formulas:

sin 𝛼𝛼 sin 𝛽𝛽 =

[

cos

− cos(𝛼𝛼 + 𝛽𝛽)

]

cos 𝛼𝛼 cos 𝛽𝛽 =

[

cos

  • cos(𝛼𝛼 + 𝛽𝛽)

]

sin 𝛼𝛼 cos 𝛽𝛽 =

[sin(𝛼𝛼 + 𝛽𝛽) + sin(𝛼𝛼 − 𝛽𝛽)]

cos 𝛼𝛼 sin 𝛽𝛽 =

[sin(𝛼𝛼 + 𝛽𝛽) − sin(𝛼𝛼 − 𝛽𝛽)]

Sum to Product Formulas:

sin 𝛼𝛼 + sin 𝛽𝛽 = 2 sin �

� cos �

sin 𝛼𝛼 − sin 𝛽𝛽 = 2 cos �

� sin �

cos 𝛼𝛼 + cos 𝛽𝛽 = 2 cos �

� cos �

cos 𝛼𝛼 − cos 𝛽𝛽 = − 2 sin �

� sin �

Sum and Difference Formulas:

sin(𝛼𝛼 ± 𝛽𝛽) = sin 𝛼𝛼 cos 𝛽𝛽 ± sin 𝛽𝛽 cos 𝛼𝛼

cos(𝛼𝛼 ± 𝛽𝛽) = cos 𝛼𝛼 cos 𝛽𝛽 ∓ sin 𝛼𝛼 sin 𝛽𝛽

tan(𝛼𝛼 ± 𝛽𝛽) =

tan 𝛼𝛼 ± tan 𝛽𝛽

1 ∓ tan 𝛼𝛼 tan 𝛽𝛽

Periodic Formulas:

sin

= sin 𝜃𝜃 csc

= csc 𝜃𝜃

cos(𝜃𝜃 + 2 𝜋𝜋𝜋𝜋) = cos 𝜃𝜃 sec(𝜃𝜃 + 2 𝜋𝜋𝜋𝜋) = sec 𝜃𝜃

tan

= tan 𝜃𝜃 cot

= cot 𝜃𝜃

Half-Angle Formulas:

sin �

1 − cos 𝜃𝜃

cos �

1 + cos 𝜃𝜃

tan �

1 − cos 𝜃𝜃

1 + cos 𝜃𝜃

Double Angle Formulas:

sin

= 2 sin 𝜃𝜃 cos 𝜃𝜃

cos( 2 𝜃𝜃) = cos

2

𝜃𝜃 −sin

2

= 2 cos

2

= 1 − 2 sin

2

tan 2 𝜃𝜃 =

2tan 𝜃𝜃

1 − tan

2

Updated: October 2019

x

y

(𝑥𝑥, 𝑦𝑦)

r θ

Trigonometric Functions:

Right Triangle: Unit Circle:

sin θ =

opposite

hypotnuse

csc θ =

hypotnuse

opposite

cos θ =

adjacent

hypotnuse

sec θ =

hypotnuse

adjacent

tan θ =

opposite

adjacent

cot θ =

adjacent

opposite

sin 𝜃𝜃 =

csc 𝜃𝜃 =

cos 𝜃𝜃 =

sec 𝜃𝜃 =

tan 𝜃𝜃 =

cot 𝜃𝜃 =

hypotenuse

opposite

adjacent

Law of Sines:

sin 𝛼𝛼

=

sin 𝛽𝛽

=

sin 𝛾𝛾

Law of Cosines:

2

2

2

− 2 𝑏𝑏𝑐𝑐 cos 𝛼𝛼

2

2

2

− 2 𝑎𝑎𝑐𝑐 cos 𝛽𝛽

2

2

− 2 𝑎𝑎𝑏𝑏 cos 𝛾𝛾

Law of Tangents:

tan

tan

tan

tan

tan

tan

Inverse Trigonometric Functions:

Definition:

𝑦𝑦 = sin

− 1

same as

����� 𝑥𝑥 = sin 𝑦𝑦

𝑦𝑦 = cos

− 1

same as

����� 𝑥𝑥 = cos 𝑦𝑦

𝑦𝑦 = tan

− 1

𝑠𝑠𝑎𝑎𝑠𝑠𝑠𝑠 𝑎𝑎𝑠𝑠

������ 𝑥𝑥 = tan 𝑦𝑦

Alternative Definition:

sin

− 1

𝑥𝑥 =arcsin 𝑥𝑥

cos

− 1

𝑥𝑥 =arccos 𝑥𝑥

tan

− 1

𝑥𝑥 =arctan 𝑥𝑥

Domain and Range:

Function Domain Range

𝑦𝑦 = sin

− 1

𝑦𝑦 = cos

− 1

𝑦𝑦 = tan

− 1

𝑦𝑦 = cot

− 1

𝑦𝑦 = sec

− 1

𝑦𝑦 = csc

− 1

Inverse Properties:

sin (sin

− 1

(𝑥𝑥)) = 𝑥𝑥 sin

− 1

(sin(𝜃𝜃)) = 𝜃𝜃

cos (cos

− 1

(𝑥𝑥)) = 𝑥𝑥 cos

− 1

(cos(𝜃𝜃)) = 𝜃𝜃

tan (tan

− 1

(𝑥𝑥)) = 𝑥𝑥 tan

− 1

(tan(𝜃𝜃)) = 𝜃𝜃