Trigonometry formula Guide GG27

7/30/2019 Trigonometry formula Guide GG27

1/23

Formulas For Triangles

#27 of Gottschalks Gestalts

A Series Illustrating Innovative Formsof the Organization & Expositionof Mathematicsby Walter Gottschalk

Infinite Vistas Press

PVD RI2001

GG27-1 (23)


2/23

2001 Walter Gottschalk

500 Angell St #414

Providence RI 02906

permission is granted without charge

to reproduce & distribute this item at cost

for educational purposes; attribution requested;no warranty of infallibility is posited

GG27-2


3/23

laws governing side / angle relationships

of triangles

consider any triangle

with vertices / angles A, B, C,

with opposite sides a, b, c,

with circumradius R;

then the following laws hold:

law of sines

= = =

D

Da

A

b

B

c

C Rsin sin sin 2

GG27-3


4/23

D

+ -

+ -

+ -

D

+ -

+( ) -

-( ) +

law of cosines

=

=

c =

variations on the law of cosines

=

=

=

cyclicly

2

a b c b c A

b c a c a B

a b a b C

a

b c b c A

b c b c

A

b c b cA

2 2 2

2 2 2

2 2

2

2 2

2 2

2 2

2

2

2

2

4 2

42

cos

cos

cos

cos

cos

sin

&

GG27-4


5/23

D+

+

+

D

- -

law of cosines - projection

=

b =

=

variations on the law of cosines - projection

= =

cyclicly

a b C c B

c A a C

c a B b A

Aa B

c a B

a C

b a C

cos cos

cos cos

cos cos

tansin

cos

sin

cos

&

GG27-5


6/23

D

-

+

-( )

+( )

-+

-( )

+( )

- -( )

+( )

D

( ) +( ) -( ) -( )

+( )

law of tangents

=

=

c a

c + a=

law of cotangents

a + b =

a b

a b

A B

A B

b c

b c

B C

B C

C A

C A

A B a b A B

b c

tan

tan

tan

tan

tan

tan

cot cot

cot

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

11

2

1

2

1

2

1

2

B C b c B C

c a C A c a C A

+( ) -( ) -( )

+( ) +( ) -( ) -( )

=

=

cot

cot cot

GG27-6


7/23

D

+( ) +( ) -( )

+( ) +( ) -( )

+( ) +( ) -( )

D

+( ) -( ) -( )

+( ) -

law of secants

=

b =

c =

law of cosecants

=

b =

a B C b c B C

C A c a C A

A B a b A B

a B C b c B C

C A c a

sec sec

sec sec

sec sec

csc csc

csc

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2(( ) -( )

+( ) -( ) -( )

csc

csc csc

1

1

2

1

2

2C A

c A B a b A B=

GG27-7


8/23

D

+ -( )

-( )

-( )

Mollweide' s equations

which are slight variations of

the law of secants

&

the law of cosecants

=

b

c + a=

c

a + b=

a

b c

A

B C

B

C A

C

A B

sin

cos

sin

cos

sin

cos

1

2

12

1

21

2

1

21

2

GG27-8


9/23

cos

sin

cos

sin

sin

a

b c=

b

c a=

=

cos1

2

- -( )

- -( )

- -( )

1

21

2

1

21

2

1

2

A

B C

B

C A

c

a b

C

A B

GG27-9


10/23

formulas for the trig fcns

of the angles of a triangle

ito the sides

the 6 basic trig fcns of the 3 angles of a triangle



with semiperimeter s

are given ito the sides by 18 formulas

of which the first 3 are listed below,

the remaining 15 then being easy consequences:

=

=

tan =

D

-( ) -( ) -( )

+ -

-( ) -( ) -( )+ -

sin

cos

A b c s s a s b s c

Ab c a

b c

As s a s b s c

b c a

2

2

4

2 2 2

2 2 2

GG27-10


11/23

D

+ -( )

--( ) -( )

also note the formulas

1 =

cyclicly

1 =

cyclicly

cos

&

cos

&

As s a

b c

As b s c

b c

2

2

GG27-11


12/23

formulas for the trig fcns

of the half - angles of a triangle

ito the sides

the 6 basic trig fcns of the 3 half - angles of a triangle



with semiperimeter s

are given ito the sides by 18 formulas

of which the first 3 are listed below,

the remaining 15 then being easy consequences:

sinA

2 =

cosA

2=

tan =

D

-( ) -( )

-( )

-( ) -( )-( )

s b s c

b c

s s a

bc

A s b s c

s s a

2

GG27-12


13/23

formulas for the area of a triangle

consider any triangle



with altitudes ha hb hc to sides a, b, c,

semiperimeter s,

with circumradius R,with inradius r,

with exradii ra rb rc to sides a, b, c;

then

the area K of the triangle

is given bythe following formulas:

D

, ,

, ,

with

GG27-13


14/23

K = & cyclicly

K = & cyclicly

K =1

2& cyclicly

K =1

2& cyclicly

K =

K = R s

K =

K =

1

2

1

2

2

42 2 2

42 2 2

2 2

2

2

2

2

a h

b c A

aB C

A

aB C

B C

R A B C

A B C

R rA B C

sA B C

a

sin

sin sin

sin

sin sin

sin

sin sin sin

sin sin sin

cos cos cos

tan tan tan

+( )

22GG27-14


15/23

K = & cyclically

K = (Heron' s formula;

already known to Archimedes)

K =

K =

K = r & cyclically

K = r r

a

a

s s aA

s s a s b s c

a b c

R

r s

s a

r rb c

( ) tan

( )( ) ( )

( )

-

- - -

-

2

4

GG27-15


16/23

D

-

many pretty formulas

are easy consequences

of the above area formulas

eg

r =

s = 4 R

=

tanA

2= = cyclicly

abc =

sin sin sin

cos cos cos

tan tan tan

&

42 2 2

2 2 2

2 2 2

4

RA B C

A B C

r sA B C

r

s a

r

s

Rrs

a

GG27-16


17/23

some trig eqns that are true

for all angles A, B, C

such that their sum is a straight angle:

A + B + C = 1 8 0

of any triangle

o = p r

hence these formulas hold

for the angles A B C

;

, ,

GG27-17


18/23

cos cos cos sin sin sin

sin sin sin sin sin sin

cos cos cos cos cos cos

sin sin sin sin sin

sin A + sin B + sin C = 4 cosA

2cos

B

2cos

C

2

=

=

= 0

A B CA B C

A B C A B C

A B C A B C

A B C A

+ + +

+ +

+ + + +

+ + +

1 42 2 2

2 2 2 4

2 2 2 4 1

4 4 4 4 2 2BB C

A B C A B C

sin

cos cos cos cos cos cos

2

4 4 4 1 4 2 2 2

= 0

=+ + +

GG27-18


19/23

tan A + tan B + tan C = tan A tan Btan C

= 1

=

=

= 1

tan

cot cot cot cot cot cot



tan tan tan tan tan tan

A B B C C A

A B C A B C

A B C A B C

A B B C C A

+ +

+ +( ) + + +

+ +

+ +

2 2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

AA

2 = tan

2

+ +

+ + +tan tan tan tan

B C A B C

2 2 2 2 2 2

22 2

GG27-19


20/23

sin sin sin cos cos cos

sin sin sin sin sin cos

sin sin sin sin cos sin

sin sin sin cos sin sin

cos cos

=

=

=

=

cos =2

2 2 2

2 2 2

2 2 2

2 2 2

2 2

2 2

2

2

2

1 2

A B C A B C

A B C A B C

A B C A B C

A B C A B C

A B C

+ + +

+ -- +

- + +

+ + - coscos cos cos

cos cos sin sin cos

cos cos sin cos sin

cos cos cos sin sin

A B C

A B C A B C

A B C A B C

A B C A B C

cos =

cos =

cos =

2

2

2

+ - -

- + -

- + + -

2 2

2 2

2 2

1 2

1 2

1 2

GG27-20


21/23

Metatheorem.

the formula F ( A, B, C ) is valid

the formula F is valid

& cyclicly

the formula F ( valid

& cyclicly

the formula is valid

the formula is valid

& cyclicly

the formula F

fi

- p +p

+p

- p - p -

p - p - p -

- p - p( )

p-

p-

p-

A B C

A B C is

F A B C

F A B C

A B C

, ,

, , )

( , , )

, ,

, ,

2 2

2 2 2

3 3 3

2 2 2 2 2 2

is valid

etc

GG27-21


22/23

T

A B C

.

sin sin sin

if

A, B, C are the angles of a triangle

then

= 0

ABC rt

4 4 4+ +

D

GG27-22


23/23

D bioline

Archimedes of Syracuse

ca 287-212 BCE

Greekmathematician, physicist, inventor;

considered to be one

of the three greatest mathematicians of all time,

the other two being Newton and Gauss

D biolineHeron of Alexandriafl 62 CE

Greek

mathematician, physicist, engineer

Dbioline

Karl Brandon Mollweide

1774-1825

German

mathematician, astronomer, cartographer

GG27-23

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Trigonometry formula Guide GG27