Download App

Sine And Cosine Formulae And Their Applications — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSine And Cosine Formulae And Their Applications5 Marks
Question
$\text{a}\sin\frac{\text{A}}{2}\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\frac{\text{B}}{2}\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)+\text{c}\sin\frac{\text{C}}{2}\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)=0.$
✓

Answer

$\text{LHS}=\text{a}\sin\frac{\text{A}}{2}\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\frac{\text{B}}{2}\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)\\+\text{c}\sin\frac{\text{C}}{2}\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)$
We know $\text{a}\sin\text{B = b}\sin\text{A,c}\sin\text{B = b}\sin\text{C},\\\text{a}\sin\text{C}=\text{c}\sin\text{B}$
$\text{a}\sin\frac{\text{A}}{2}\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\frac{\text{B}}{2}\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)\\+\text{c}\sin\frac{\text{C}}{2}\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)=0$
$=\text{a}\sin\Big(\frac{\pi-(\text{B + C})}{2}\Big)\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\Big(\frac{\pi-(\text{C + A})}{2}\Big)\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)\\+\text{c}\sin\Big(\frac{\pi-(\text{A + B})}{2}\Big)\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)$
$=\text{a}\cos\Big(\frac{\text{B + C}}{2}\Big)\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\cos\Big(\frac{\text{C + A}}{2}\Big)\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)\\+\text{c}\cos\Big(\frac{\text{A + B}}{2}\Big)\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)$
$=\text{a}(\sin\text{B}-\sin\text{C})+\text{b}(\sin\text{C}-\sin\text{A})+\text{c}(\sin\text{A}-\sin\text{B})$
$=\text{a}\sin\text{B}-\text{a}\sin\text{C}+\text{b}\sin\text{C}-\text{b}\sin\text{A + c}\sin\text{A}-\text{c}\sin\text{B}$
$=\text{b}\sin\text{A}-\text{a}\sin\text{C + b}\sin\text{C}-\text{b}\sin\text{A + a}\sin\text{C}-\text{b}\sin\text{C}$
$=0 =\text{RHS}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Sine And Cosine Formulae And Their Applications→Sine And Cosine Formulae And Their Applications — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the equations of the sides of the triangles the coordinates of whose angular points are respectively:
(0, 1), (2, 0) and (-1, -2).
→
Find the equation of the parabola, if
The focus is at $(a, 0)$ and the vertex is at $(a', 0).$
→
Sum the following series to n terms:
$1 + 3 + 6 + 10 + 15 + .....$
→
Show that the plane $ax+ by+ cz + d = 0$ divides the line joining the points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the ratio $=\frac{\text{ax}_1+\text{by}_1+\text{cz}_1+d}{\text{ax}_2+\text{by}_2+\text{cz}_2+d}.$
→
Sketch the graphs of the following functions:
$\text{f(x)}=\cot2\text{x}$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{9}-\text{a}^9}{\text{x}-\text{a}}=9,$ find all possible value of a.
→
Find the distance of the point of intersection of the lines $2x + 3y = 21$ and $3x - 4y + 11 = 0$ from the line $8x + 6y + 5 = 0$.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\tan3\text{x}-2\text{x}}{3\text{x}-\sin^2\text{x}}$
→
Find the point to which the origin should be shifted after a translation of axes so that the following equations will have no first deree terms:
$x^2 + y^2 - 5x + 2y - 5 = 0$
→
The distributive law from algebra states that for all real numbers $c, a_1$ and $a_2$, we have $c(a_1 + a_2)=ca_1 + ca_2.$
Use this law and mathematical induction to prove that, for all natural numbers $\text{n}\geq2,$ if $c, a_1, a_2, ...,$ an are any real numbers, then $c(a_1 + a_2 +...+ a_{n)}= ca_1 + ca_2 +...+ ca_n.$
→