Question

In a∆ABC, a, c, A are given and b₁, b₂ are two values of the third side b such that b₂ = 2b₁. Then sinA is equal to

• A. $\sqrt { \frac { 9 a ^ { 2 } - c ^ { 2 } } { 8 a ^ { 2 } } }$
• B. $\sqrt { \frac { 9 a ^ { 2 } - c ^ { 2 } } { 8 c ^ { 2 } } }$
• C. $\sqrt { \frac { 9 a ^ { 2 } - c ^ { 2 } } { 8 a ^ { 2 } } }$
• D. None of these

Answer 1

Answer: (B)

$\sqrt { \frac { 9 a ^ { 2 } - c ^ { 2 } } { 8 c ^ { 2 } } }$

Answer 2

We have cos A =

⇒ b² – 2bc cos A + (c² – a²) = 0.

it is given that b₁ and b₂ are roots of this equation.

Therefore b₁ + b₂ = 2ccosA and b₁b₂ = c² – a² ⇒ 3b₁

= 2c cos A and 2b₁² = c² – a² (since b₂ = 2b₁ given)

⇒ 2 $\left( \frac { 2 c } { 3 } \cos A \right) ^ { 2 } = c ^ { 2 } - a ^ { 2 }$

⇒ 8c²(1 – sin²A) = 9c² – 9a² ⇒ sin A =$\sqrt { \frac { 9 a ^ { 2 } - c ^ { 2 } } { 8 c ^ { 2 } } }$