MCQ
If $A, B$ and $C$ are interior angles of a triangle $\text{ABC,}$ then $\sin\Big(\frac{\text{B}+\text{C}}{2}\Big)=$
- A$\sin\frac{\text{A}}{2}$
- ✓$\cos\frac{\text{A}}{2}$
- C$-\sin\frac{\text{A}}{2}$
- D$-\cos\frac{\text{A}}{2}$
✓
Answer
Correct option: B.
$\cos\frac{\text{A}}{2}$
We know tht in triangle $\text{ABC}$
$\text{A+B+C}=180^\circ$
$\Rightarrow\text{B+C}=180^\circ-\text{A}$
$\Rightarrow\frac{\text{B+C}}{2}=\frac{90^\circ}{2}-\frac{\text{A}}{2}$
$\Rightarrow\sin\Big(\frac{\text{B+C}}{2}\Big)=\sin\Big(90^\circ-\frac{\text{A}}{2}\Big)$
Since $\sin(90^\circ-\text{A})=\cos{\text{A}}$
So,
$\Rightarrow\sin\Big(\frac{\text{B+C}}{2}\Big)=\cos\frac{\text{A}}{2}$
Hence the correct option is $(b)$
$\text{A+B+C}=180^\circ$
$\Rightarrow\text{B+C}=180^\circ-\text{A}$
$\Rightarrow\frac{\text{B+C}}{2}=\frac{90^\circ}{2}-\frac{\text{A}}{2}$
$\Rightarrow\sin\Big(\frac{\text{B+C}}{2}\Big)=\sin\Big(90^\circ-\frac{\text{A}}{2}\Big)$
Since $\sin(90^\circ-\text{A})=\cos{\text{A}}$
So,
$\Rightarrow\sin\Big(\frac{\text{B+C}}{2}\Big)=\cos\frac{\text{A}}{2}$
Hence the correct option is $(b)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In step deviation method, $u$ is given by
→cosec45° = ?
If each edge of a cube is increased by $50 \%$, the percentage increase in the surface area is:
→→If $\alpha,\ \beta$ are the zeros of the polynomial $x^2 + 6x + 2$, then $\Big(\frac{1}{\alpha}+\frac{1}{\beta}\Big)=?$
→In an A.P., $a=8, t_{ n }=62, S_{ n }=210$ then find n .
→Two coins are tossed simultaneously. What is the probability of getting at most one head?
If $2x - 3y = 7$ and $(a + b)x - (a + b - 3)y = 4a + b$ have an infinite number of solutions, then:
→In the given figure, $\text{QR}$ is a common tangent to the given circles touching externally at the point $T.$ The tangent at $T$ meets $\text{QR}$ at $P.$ If $\text{PT} = 3.8\ cm,$ then the length of $\text{QR} ($ in $cm)$ is:
→If the mode of the data: $64, 60, 48, x, 43, 48, 43, 34$ is $43,$ then $x + 3 =$
→