Answer
Verified
39k+ views
Hint:
In this case, we will use the angle sum property of a triangle according to which the triangle's triangle's interior angles add up to $180^{o}$. Using this relation and putting it in the given scenario we will get the result.
Formula Used:
Three sides in the triangle $ABC$ are represented by the letters $AB, BC,$ and $CA$. The three vertices are $A, B$, and $C$, and the three interior angles are $\angle ABC, \angle BCA,$ and $\angle CAB$. Then according to the angle sum property of the triangle: $\angle ACB + \angle BAC + \angle CBA= 180^{o}$
Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$b^2 = a^2 +c^2 − 2ac.\cos B$.
Complete step-by-step solution:
We have, $2a\sin (\dfrac{A-B+C}{2})$
Since $ABC$ is a triangle, therefore we can say;
$A+B+C=180^{o}\\
A+C=180^{o}-B …(i)$
Now consider
$2a\sin (\dfrac{A-B+C}{2})$
Substitute value of $A+C$ using eq (i)
$\Rightarrow2a\sin \,\,\left( \dfrac{180^{o}-B-B}{2} \right)\\
\Rightarrow2a\sin(\dfrac{180^{o}-2B}{2})\\
\Rightarrow 2ac \sin(90^{o}-B)
\Rightarrow 2ac \cos B$
Using the cosine rule of the triangle we get;
$\therefore b^2 = a^2 +c^2 − 2ac.\cos B\\
\therefore \cos B=\dfrac{a^2+c^2-b^2}{2ac}\\
\Rightarrow 2ac \dfrac{a^2+c^2-b^2}{2ac}\\
\Rightarrow a^2+c^2-b^2$
$2a\sin \,\,\left( \dfrac{A-B+C}{2} \right)=a^2+c^2-b^2$.
Hence, $2a\sin \,\,\left( \dfrac{A-B+C}{2} \right)=a^2+c^2-b^2$. So, option B is correct .
Note:
Keep in mind that such type of question requires knowledge of the basic trigonometric conversation. Here we have uses the conversion of sine at $(90^{o}-B)$ which turns out to be $\cos B$ as it lies in the first quadrant so it is taken as positive. Using the law of cosine for angle B, students usually make mistakes in taking the wrong angles. Different angles use cosine rules in different ways.
In this case, we will use the angle sum property of a triangle according to which the triangle's triangle's interior angles add up to $180^{o}$. Using this relation and putting it in the given scenario we will get the result.
Formula Used:
Three sides in the triangle $ABC$ are represented by the letters $AB, BC,$ and $CA$. The three vertices are $A, B$, and $C$, and the three interior angles are $\angle ABC, \angle BCA,$ and $\angle CAB$. Then according to the angle sum property of the triangle: $\angle ACB + \angle BAC + \angle CBA= 180^{o}$
Laws of cosine for triangle $ABC$ whose length is $a, b$, and $c$ respectively is given by;
$b^2 = a^2 +c^2 − 2ac.\cos B$.
Complete step-by-step solution:
We have, $2a\sin (\dfrac{A-B+C}{2})$
Since $ABC$ is a triangle, therefore we can say;
$A+B+C=180^{o}\\
A+C=180^{o}-B …(i)$
Now consider
$2a\sin (\dfrac{A-B+C}{2})$
Substitute value of $A+C$ using eq (i)
$\Rightarrow2a\sin \,\,\left( \dfrac{180^{o}-B-B}{2} \right)\\
\Rightarrow2a\sin(\dfrac{180^{o}-2B}{2})\\
\Rightarrow 2ac \sin(90^{o}-B)
\Rightarrow 2ac \cos B$
Using the cosine rule of the triangle we get;
$\therefore b^2 = a^2 +c^2 − 2ac.\cos B\\
\therefore \cos B=\dfrac{a^2+c^2-b^2}{2ac}\\
\Rightarrow 2ac \dfrac{a^2+c^2-b^2}{2ac}\\
\Rightarrow a^2+c^2-b^2$
$2a\sin \,\,\left( \dfrac{A-B+C}{2} \right)=a^2+c^2-b^2$.
Hence, $2a\sin \,\,\left( \dfrac{A-B+C}{2} \right)=a^2+c^2-b^2$. So, option B is correct .
Note:
Keep in mind that such type of question requires knowledge of the basic trigonometric conversation. Here we have uses the conversion of sine at $(90^{o}-B)$ which turns out to be $\cos B$ as it lies in the first quadrant so it is taken as positive. Using the law of cosine for angle B, students usually make mistakes in taking the wrong angles. Different angles use cosine rules in different ways.
Recently Updated Pages
Let gx 1 + x x and fx left beginarray20c 1x 0 0x 0 class 12 maths JEE_Main
The number of ways in which 5 boys and 3 girls can-class-12-maths-JEE_Main
Find dfracddxleft left sin x rightlog x right A left class 12 maths JEE_Main
Distance of the point x1y1z1from the line fracx x2l class 12 maths JEE_Main
In a box containing 100 eggs 10 eggs are rotten What class 12 maths JEE_Main
dfracddxex + 3log x A ex cdot x2x + 3 B ex cdot xx class 12 maths JEE_Main
Other Pages
The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main
What is the volume of water that must be added to a class 11 chemistry JEE_Main
Oxidation state of S in H2S2O8 is A 6 B 7 C +8 D 0 class 12 chemistry JEE_Main
A closed organ pipe and an open organ pipe are tuned class 11 physics JEE_Main
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
When an electron and a proton are placed in an electric class 12 physics JEE_Main