Question

In $△ABC$, if $a,c,A$ are given and $b_1,b_2$ are two values of third side $b$ such that $b_2=2b_1$. Then, $\sin A$ is equal to

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

Answer 1

Answer: (B)

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

We have $\cos A=\frac{b^2+c^2-a^2}{2bc}$

$b^2-2bc\cos A+(c^2-a^2)=0$

Since $b_1$ and $b_2$ are in the roots of this equation,

thus $b_1+b_2=2c\cos A$

$b_1{b}_2=c^2-a^2$

$\Rightarrow 3b_1=2c\cos A;2b_1^2=c^2-a^2[\because b_2=2b_1]$

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

$\Rightarrow 8c^2(1-{\sin }^{2}A)=9c^2-9a^2$

Therefore, $\sin A=\sqrt{\frac{9a^2-c^2}{8c^2}}$