Question

Express each of the following as an algebraic sum of sines or cosines :

(i) 2 sin $3x$ cos $2x$

(ii) 2 cos $4x$ sin 2$x$

(iii) 2 cos $6x$ cos $4x$

(iv) 2 sin $3x$ sin $5x$

Answer 1

(i) We know that 2 sin A cos B = sin (A + B) + sin (A-B).

$\therefore$ 2 sin $3x$ cos $2x$ = sin $(3x+2x)$ + sin $(3x-2x)$

= sin $5x$+sin $x$.

(ii) We know that 2 cos A Sin B = sin (A + B) - sin (A-B)

$\therefore$ 2 cos $4x$ sin 2$x$ = sin $(4x+2x)$-sin$(4x-2x)$

= sin $6x$ - sin $2x$

(iii) We know that 2 cos A cos B = cos (A + B) + cos (A-B).

$\therefore$ 2 cos $6x$ cos $4x$ = cos $(6x+4x)$ + cos $(6x-4x)$

= cos $10x$ + cos $2x$

(iv) We know that 2 sin A sin B = cos (A-B) - cos (A + B)

$\therefore$ 2 sin $3x$ sin $5x$

= cos $(5x-3x)$ - cos $(5x+3x)$

= cos $2x$ - cos $8x$