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