prove-sin-a-b-sin-a-cos-b-sin-b-cos-a-

Question Number 216202 by mahdipoor last updated on 29/Jan/25

p r o v e :

s i n (a + b) = s i n (a) c o s (b) + s i n (b) c o s (a)

Answered by dionigi last updated on 30/Jan/25

e^{i (a + b)} = \cos (a + b) + i \sin (a + b)

e^{i (a + b)} = e^{i a} e^{i b}

e^{i (a + b)} = (\cos a + i \sin a) (\cos b + i \sin b)

e^{i (a + b)} = \cos a \cos b - \sin a \sin b

+ i \sin a \cos b + i \cos a \sin b

s p l i t i n g r e a l a n d i m a g i n a r y p a r t s

o f f i r s t a n d l a s t l i n e

\cos (a + b) = \cos a \cos b - \sin a \sin b

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

Answered by mehdee7396 last updated on 30/Jan/25

E C = E F \times C o s α & E B = A E \times S i n α

S i n (α + β) = \frac{A D}{A F} = \frac{E C}{A F} + \frac{B E}{A F}

= \frac{E F}{A F} \times C o s α + \frac{A E}{A F} \times S i n α

= S i n β C o s α + S i n α C o s β

Commented by mehdee7396 last updated on 30/Jan/25

Answered by a.lgnaoui last updated on 30/Jan/25

Soit OB : Cos (a + b) i, (Sin (a + b) j

C alcul de OB / Repere (O, A, A^{'})

ci - joint demonstration :

Answered by a.lgnaoui last updated on 30/Jan/25