Trigonometry Sum Angle Identities

by: javier

The MOTHER OF THEM ALL

The MOTHER OF THEM ALL

the M.O.T.A. 7−→

cos(a − b) = cos a cos b + sin a sin b cos(a + b) = cos a cos b − sin a sin b sin(a + b) = sin a cos b + cos a sin b

sin(a − b) = sin a cos b − cos a sin b tan a + tan b tan a − tan b tan(a + b) = tan(a − b) = 1 − tan a tan b 1 + tan a tan b

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

substitute a = x and b = −y

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

substitute a =

π 2

− x and b = −y

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

The MOTHER OF THEM ALL cos(a − b) = cos a cos b + sin a sin b

substitute a = x and b = −x

The MOTHER OF THEM ALL sin(a + b) = sin a cos b + cos a sin b

The MOTHER OF THEM ALL sin(a + b) = sin a cos b + cos a sin b

substitute a = x and b = x

The MOTHER OF THEM ALL sin(a + b) = sin a cos b + cos a sin b sin(a − b) = sin a cos b − cos a sin b

The MOTHER OF THEM ALL sin(a + b) = sin a cos b + cos a sin b sin(a − b) = sin a cos b − cos a sin b

Add them

The MOTHER OF THEM ALL eix = cos(x) + i sin(x)

The MOTHER OF THEM ALL

eix = cos(x) + i sin(x)

e−ix = cos(x) − i sin(x)

Add them

THE PROOF

THE PROOF

11

b

a

THE PROOF

11

b

a

THE PROOF

11

b

a

THE PROOF

11

b

a