Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If $a=16,b=24,c=20$ then $\cos \left(\frac{B}{2}\right)=?$

• A. $\frac{3}{4}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

Answer: (A)

$\frac{3}{4}$

From the cosine rule, we have

$\cos \left(B\right)=\frac{a^2+c^2-b^2}{2ac}$

$\cos \left(B\right)=\frac{256+400-576}{2\times 16\times 20}=\frac{80}{640}=\frac{1}{8}$

Now,

$\cos \frac{B}{2}=\sqrt{\frac{1+\cos \left(B\right)}{2}}$

$\Rightarrow \cos \frac{B}{2}=\sqrt{\frac{1+\frac{1}{8}}{2}}$

$\Rightarrow \cos \frac{B}{2}=\sqrt{\frac{9}{16}}$

$\Rightarrow \cos \frac{B}{2}=\frac{3}{4}$