If $A , B$ and $C$ are interior angles of a triangle $ABC ,$ then show that
$\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$
If $A , B$ and $C$ are interior angles of a triangle $ABC ,$ then show that
$\sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2}$
We know that for a triangle $ABC$
$\angle A+\angle B+\angle C=180^{\circ}$
$\angle B+\angle C=180^{\circ}-\angle A$
$\frac{\angle B+\angle C}{2}=90^{\circ}-\frac{\angle A}{2}$
$\sin \left(\frac{B+C}{2}\right)=\sin \left(90^{\circ}-\frac{A}{2}\right)$
$=\cos \left(\frac{ A }{2}\right)$
Similar Questions
State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$
$(ii)$ cot $A$ is the product of cot and $A$.
$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.
State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$
$(ii)$ cot $A$ is the product of cot and $A$.
$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.
If $\sin A =\frac{3}{4},$ calculate $\cos A$ and $\tan A$.
If $\sin A =\frac{3}{4},$ calculate $\cos A$ and $\tan A$.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2 \sec A$
Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:
$(i)$ $\sin A \cos C+\cos A \sin C$
$(ii)$ $\cos A \cos C-\sin A \sin C$
In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:
$(i)$ $\sin A \cos C+\cos A \sin C$
$(ii)$ $\cos A \cos C-\sin A \sin C$