goniometric formulas

goniometric formulas

The

goniometric

(from Greek

γ

⁢

ω

⁢

ν

í

α

“angle” and

μ

⁢

ε

⁢

τ

⁢

ϱ

⁢

ι

⁢

κ

ó

ς

“measuring”) concerns the

trigonometric functions

and their mutual connections. There are a great amount of formulas involving these

functions

(usually for real

arguments

).

1.

Pythagorean identities

–

sin

2

⁡

x

+

cos

2

⁡

x

=

1

–

tan

2

⁡

x

+

1

=

sec

2

⁡

x

–

1

+

cot

2

⁡

x

=

csc

2

⁡

x

2.

Fractional

identities

–

tan

⁡

x

=

sin

⁡

x

cos

⁡

x

–

cot

⁡

x

=

cos

⁡

x

sin

⁡

x

–

cot

⁡

x

=

1

tan

⁡

x

–

tan

⁡

x

=

1

cot

⁡

x

–

csc

⁡

x

=

1

sin

⁡

x

–

sec

⁡

x

=

1

cos

⁡

x

3.

Formulas involving

radicals

(

http://planetmath.org/Radical6

)

–

sin

⁡

x

=

±

tan

⁡

x

1

+

tan

2

⁡

x

–

cos

⁡

x

=

±

1

1

+

tan

2

⁡

x

4.

Weierstrass substitution formulas

and related formula for

tan

⁡

x

–

sin

⁡

x

=

2

⁢

tan

⁡

(

x

2

)

1

+

tan

2

⁡

(

x

2

)

–

cos

⁡

x

=

1

-

tan

2

⁡

(

x

2

)

1

+

tan

2

⁡

(

x

2

)

–

tan

⁡

x

=

2

⁢

tan

⁡

(

x

2

)

1

-

tan

2

⁡

(

x

2

)

5.

Trigonometric functions of a

purely imaginary number

–

sin

⁡

(

i

⁢

x

)

=

i

⁢

sinh

⁡

x

–

cos

⁡

(

i

⁢

x

)

=

cosh

⁡

x

–

tan

⁡

(

i

⁢

x

)

=

i

⁢

tanh

⁡

x

–

cot

⁡

(

i

⁢

x

)

=

i

⁢

coth

⁡

x

–

csc

⁡

(

i

⁢

x

)

=

i

⁢

csch

⁡

x

–

sec

⁡

(

i

⁢

x

)

=

sech

⁡

x

6.

Addition formulas

and subtraction formulas (

http://planetmath.org/AdditionFormulasForSineAndCosine

)

–

sin

⁡

(

x

±

y

)

=

sin

⁡

x

⁢

cos

⁡

y

±

cos

⁡

x

⁢

sin

⁡

y

–

cos

⁡

(

x

±

y

)

=

cos

⁡

x

⁢

cos

⁡

y

∓

sin

⁡

x

⁢

sin

⁡

y

–

tan

⁡

(

x

±

y

)

=

tan

⁡

x

±

tan

⁡

y

1

∓

tan

⁡

x

⁢

tan

⁡

y

7.

Formulas for trigonometric functions of a

complex number

–

sin

⁡

(

x

+

i

⁢

y

)

=

sin

⁡

x

⁢

cosh

⁡

y

+

i

⁢

cos

⁡

x

⁢

sinh

⁡

y

–

cos

⁡

(

x

+

i

⁢

y

)

=

cos

⁡

x

⁢

cosh

⁡

y

-

i

⁢

sin

⁡

x

⁢

sinh

⁡

y

–

tan

⁡

(

x

+

i

⁢

y

)

=

tan

⁡

x

+

i

⁢

tanh

⁡

y

1

-

i

⁢

tan

⁡

x

⁢

tanh

⁡

y

8.

Complement formulas

–

sin

⁡

(

π

2

-

x

)

=

cos

⁡

x

–

cos

⁡

(

π

2

-

x

)

=

sin

⁡

x

–

tan

⁡

(

π

2

-

x

)

=

cot

⁡

x

9.

Supplement formulas

–

sin

⁡

(

π

-

x

)

=

sin

⁡

x

–

cos

⁡

(

π

-

x

)

=

-

cos

⁡

x

–

tan

⁡

(

π

-

x

)

=

-

tan

⁡

x

10.

Explement formulas

–

sin

⁡

(

2

⁢

π

-

x

)

=

-

sin

⁡

x

–

cos

⁡

(

2

⁢

π

-

x

)

=

cos

⁡

x

–

tan

⁡

(

2

⁢

π

-

x

)

=

-

tan

⁡

x

11.

angle formulas

–

sin

⁡

(

-

x

)

=

-

sin

⁡

x

–

cos

⁡

(

-

x

)

=

cos

⁡

x

–

tan

⁡

(

-

x

)

=

-

tan

⁡

x

12.

Periodicity

(

http://planetmath.org/Periodic

) formulas

–

sin

⁡

(

x

+

2

⁢

π

)

=

sin

⁡

x

–

cos

⁡

(

x

+

2

⁢

π

)

=

cos

⁡

x

–

tan

⁡

(

x

+

π

)

=

tan

⁡

x

13.

Double angle formulas

–

sin

⁡

(

2

⁢

x

)

=

2

⁢

sin

⁡

x

⁢

cos

⁡

x

–

cos

⁡

(

2

⁢

x

)

=

cos

2

⁡

x

-

sin

2

⁡

x

=

2

⁢

cos

2

⁡

x

-

1

=

1

-

2

⁢

sin

2

⁡

x

–

tan

⁡

(

2

⁢

x

)

=

2

⁢

tan

⁡

x

1

-

tan

2

⁡

x

14.

Triple angle formulas

–

sin

⁡

(

3

⁢

x

)

=

3

⁢

sin

⁡

x

-

4

⁢

sin

3

⁡

x

=

(

4

⁢

cos

2

⁡

x

-

1

)

⁢

sin

⁡

x

–

cos

⁡

(

3

⁢

x

)

=

4

⁢

cos

3

⁡

x

-

3

⁢

cos

⁡

x

=

(

1

-

4

⁢

sin

2

⁡

x

)

⁢

cos

⁡

x

–

tan

⁡

(

3

⁢

x

)

=

3

⁢

tan

⁡

x

-

tan

3

⁡

x

1

-

3

⁢

tan

2

⁡

x

15.

Half angle formulas

–

sin

⁡

(

x

2

)

=

±

1

-

cos

⁡

x

2

–

cos

⁡

(

x

2

)

=

±

1

+

cos

⁡

x

2

–

tan

⁡

(

x

2

)

=

sin

⁡

x

1

+

cos

⁡

x

=

1

-

cos

⁡

x

sin

⁡

x

=

±

1

-

cos

⁡

x

1

+

cos

⁡

x

16.

Prosthaphaeresis formulas

–

sin

⁡

x

+

sin

⁡

y

=

2

⁢

sin

⁡

(

x

+

y

2

)

⁢

cos

⁡

(

x

-

y

2

)

–

sin

⁡

x

-

sin

⁡

y

=

2

⁢

sin

⁡

(

x

-

y

2

)

⁢

cos

⁡

(

x

+

y

2

)

–

cos

⁡

x

+

cos

⁡

y

=

2

⁢

cos

⁡

(

x

+

y

2

)

⁢

cos

⁡

(

x

-

y

2

)

–

cos

⁡

x

-

cos

⁡

y

=

-

2

⁢

sin

⁡

(

x

+

y

2

)

⁢

sin

⁡

(

x

-

y

2

)

17.

formulas

–

sin

⁡

x

⁢

sin

⁡

y

=

cos

⁡

(

x

-

y

)

-

cos

⁡

(

x

+

y

)

2

–

cos

⁡

x

⁢

sin

⁡

y

=

sin

⁡

(

x

+

y

)

-

sin

⁡

(

x

-

y

)

2

–

cos

⁡

x

⁢

cos

⁡

y

=

cos

⁡

(

x

-

y

)

+

cos

⁡

(

x

+

y

)

2

18.

Other sums and differences

–

tan

⁡

x

±

tan

⁡

y

=

sin

⁡

(

x

±

y

)

cos

⁡

x

⁢

cos

⁡

y

–

cot

⁡

x

±

cot

⁡

y

=

sin

⁡

(

y

±

x

)

sin

⁡

x

⁢

sin

⁡

y

–

cos

⁡

x

±

sin

⁡

x

=

2

⁢

sin

⁡

(

π

4

±

x

)

=

2

⁢

cos

⁡

(

π

4

∓

x

)

19.

formulas

–

Second power

*

sin

2

⁡

x

=

1

-

cos

⁡

(

2

⁢

x

)

2

*

cos

2

⁡

x

=

1

+

cos

⁡

(

2

⁢

x

)

2

*

tan

2

⁡

x

=

1

-

cos

⁡

(

2

⁢

x

)

1

+

cos

⁡

(

2

⁢

x

)

–

Third power

*

sin

3

⁡

x

=

3

⁢

sin

⁡

x

-

sin

⁡

(

3

⁢

x

)

4

*

cos

3

⁡

x

=

3

⁢

cos

⁡

x

+

cos

⁡

(

3

⁢

x

)

4

*

tan

3

⁡

x

=

3

⁢

sin

⁡

x

-

sin

⁡

(

3

⁢

x

)

3

⁢

cos

⁡

x

+

cos

⁡

(

3

⁢

x

)

–

Fourth power

*

sin

4

⁡

x

=

cos

⁡

(

4

⁢

x

)

-

4

⁢

cos

⁡

(

2

⁢

x

)

+

3

8

*

cos

4

⁡

x

=

cos

⁡

(

4

⁢

x

)

+

4

⁢

cos

⁡

(

2

⁢

x

)

+

3

8

*

tan

4

⁡

x

=

cos

⁡

(

4

⁢

x

)

-

4

⁢

cos

⁡

(

2

⁢

x

)

+

3

cos

⁡

(

4

⁢

x

)

+

4

⁢

cos

⁡

(

2

⁢

x

)

+

3

20.

Recursion formulas

–

sin

⁡

[

(

n

+

1

)

⁢

x

]

=

2

⁢

cos

⁡

x

⁢

sin

⁡

(

n

⁢

x

)

-

sin

⁡

[

(

n

-

1

)

⁢

x

]

–

cos

⁡

[

(

n

+

1

)

⁢

x

]

=

2

⁢

cos

⁡

x

⁢

cos

⁡

(

n

⁢

x

)

-

cos

⁡

[

(

n

-

1

)

⁢

x

]

21.

Exponential formulas (

http://planetmath.org/ExponentialFunction

)

–

e

i

⁢

x

=

cos

⁡

x

+

i

⁢

sin

⁡

x

–

e

-

i

⁢

x

=

cos

⁡

x

-

i

⁢

sin

⁡

x

–

cos

⁡

x

=

e

i

⁢

x

+

e

-

i

⁢

x

2

–

sin

⁡

x

=

e

i

⁢

x

-

e

-

i

⁢

x

2

⁢

i

–

tan

⁡

x

=

e

i

⁢

x

-

e

-

i

⁢

x

i

⁢

(

e

i

⁢

x

+

e

-

i

⁢

x

)

22.

Some special formulas

–

tan

⁡

(

x

+

π

4

)

=

cos

⁡

x

+

sin

⁡

x

cos

⁡

x

-

sin

⁡

x

=

±

1

+

sin

⁡

2

⁢

x

1

-

sin

⁡

2

⁢

x

–

tan

⁡

x

+

sec

⁡

x

=

tan

⁡

(

x

2

+

π

4

)

–

tan

⁡

(

x

±

y

2

)

=

sin

⁡

x

±

sin

⁡

y

cos

⁡

x

+

cos

⁡

y

=

cos

⁡

y

-

cos

⁡

x

sin

⁡

x

∓

sin

⁡

y

–

tan

⁡

(

x

+

y

2

)

⁢

tan

⁡

(

x

-

y

2

)

=

cos

⁡

y

-

cos

⁡

x

cos

⁡

y

+

cos

⁡

x

–

sin

⁡

(

x

+

y

)

⁢

sin

⁡

(

x

-

y

)

=

sin

2

⁡

x

-

sin

2

⁡

y

Title

goniometric formulas

Canonical name

GoniometricFormulas

Date of creation

2013-03-22 17:00:40

Last modified on

2013-03-22 17:00:40

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

39

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 33B10

Classification

msc 26A09

Synonym

trigonometric identities

Synonym

goniometric formulae

Related topic

Trigonometry

Related topic

DefinitionsInTrigonometry

Related topic

ExampleOnSolvingAFunctionalEquation

Related topic

IntegrationOfRationalFunctionOfSineAndCosine

Related topic

WeierstrassSubstitutionFormulas

Related topic

TangentOfHalvedAngle

Related topic

ExampleOfTelescopingSum

Related topic

DerivativeForParametricForm

Related topic

ComplementaryAngles

Related topic

S

Defines

supplement formula

Defines

complement formula

Defines

half angle formula

Defines

product formula

Defines

Pythagorean identities

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