Question

In a $\Delta ABC$, a=1, c=2 and A is given. If $b_1,b_2$ are two values of b such that $2b_2=b_1$ then $sin$A=

• A. $\sqrt{\frac{5}{8}}$
• B. $\sqrt{\frac{5}{16}}$
• C. $\sqrt{\frac{5}{32}}$
• D. $\sqrt{\frac{3}{16}}$

Answer 1

Answer: (C)

$\sqrt{\frac{5}{32}}$

$\angle A$ is given

$a=1$

$c=2$

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

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

Roots are $b_1$ & $b_2$ (Given $2b_2=b_1$)

$b_1+b_2=3b_2=2c\cos A\to$ sum of roots

$b_1{b}_2=2b_2^2=c^2-a^2\to$ product of roots

$2b_2^2=2^2-1^2$

$\Rightarrow b_2^2=\frac{3}{2}$

$\Rightarrow b_2=\sqrt{\frac{3}{2}}$

$3b_2=2c\cos A$

$\Rightarrow \frac{3\sqrt{3}}{\sqrt{2}}=2\left(2\right)\cos A$

$\Rightarrow \cos A=\frac{3\sqrt{3}}{4\sqrt{2}}$

$\sin A=\sqrt{1-{\cos }^{2}A}$

$=\sqrt{1-\frac{27}{32}}$

$\sin A=\sqrt{\frac{5}{32}}$.