Question

In ΔABC, `(sin(B - C))/(sin(B + C))` = ______

Options

• `(b^2 + c^2)/(a^2)`

• `(b^2 - c^2)/(a^2)`

• `a^2/(b^2 + c^2)`

• `a^2/(b^2 - c^2)`

Answer 1

In ΔABC, `(sin(B - C))/(sin(B + C))` =  `underline((b^2 - c^2)/(a^2))`

Explanation:

`(sin(B - C))/(sin(B + C))`

= `(sinBcosC - cosBsinC)/(sin(pi - A))` .........[∵ A + B + C = π]

= `(sinBcosC - cosBsinC)/sinA`

= `b/acosC - c/acosB` ..........[by sine rule]

= `b/a((a^2 + b^2 - c^2)/(2ab)) - c/a((a^2 + c^2 - b^2)/(2ac))` ................... [by cosine rule]

= `1/(2a^2)(a^2 + b^2 - c^2 - a^2 - c^2 + b^2)`

= `(2b^2 - 2c^2)/(2a^2) = (b^2 - c^2)/a^2`