Answer

Verified

435.6k+ views

**Hint:**In order to prove the given we make use of the Lami’s theorem. Lami’s Theorem states that when three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. If we join the tails of these vectors it forms a triangle.

The Lami’s theorems:

${\text{k = }}\dfrac{{{\text{Sin A}}}}{{\text{a}}}{\text{ = }}\dfrac{{{\text{Sin B}}}}{{\text{b}}}{\text{ = }}\dfrac{{{\text{Sin C}}}}{{\text{c}}}$

(k is a constant, denoting all are equal to k)

**Complete step-by-step answer:**

Let us construct a triangle ABC with sides of lengths a, b and c respectively as shown in the figure below.

From the diagram we are very clear that A, B and C are angles of the triangle whereas a, b and c are respective sides.

We know the formula given by the Lami’s theorem, as follows:

${\text{k = }}\dfrac{{{\text{Sin A}}}}{{\text{a}}}{\text{ = }}\dfrac{{{\text{Sin B}}}}{{\text{b}}}{\text{ = }}\dfrac{{{\text{Sin C}}}}{{\text{c}}}$

(k is a constant, denoting all are equal to k)

We know,

From the formula, ${\text{a = }}\dfrac{{{\text{Sin A}}}}{{\text{k}}},{\text{b = }}\dfrac{{{\text{Sin B}}}}{{\text{k}}}{\text{ and c = }}\dfrac{{{\text{Sin C}}}}{{\text{k}}}$

We know from the identities of trigonometric ratios that, ${\text{Sin x - Sin y = 2Cos}}\left( {\dfrac{{{\text{x + y}}}}{2}} \right)\operatorname{Sin} \left( {\dfrac{{{\text{x - y}}}}{2}} \right)$

Also given angles, A, B and C form a triangle, hence A + B + C = 180°

Also according to the properties of Sin and Cos functions, Sin (180 – x) = Sin x and Cos (180 – x) = -Cos x

Using all the above relations and identities, we solve to obtain the given as follows:

$

\Rightarrow 2{\text{Cos}}\left( {\dfrac{{{\text{B + C}}}}{2}} \right){\text{Sin}}\left( {\dfrac{{{\text{B - C}}}}{2}} \right) = \left( {{\text{b - c}}} \right){\text{k}} \\

\Rightarrow {\text{2Sin}}\dfrac{{\text{A}}}{2}{\text{Sin}}\left( {\dfrac{{{\text{B - C}}}}{2}} \right) = \left( {{\text{b - c}}} \right){\text{k}} \\

\left( {\because {\text{SinA = ak}}} \right) \\

\Rightarrow 2{\text{Sin}}\dfrac{{\text{A}}}{2}{\text{cos}}\dfrac{{\text{A}}}{2} = {\text{ak}} \\

\Rightarrow \dfrac{{{\text{Sin}}\left( {\dfrac{{{\text{B - C}}}}{2}} \right)}}{{{\text{cos}}\dfrac{{\text{A}}}{2}}} = \left( {\dfrac{{{\text{b - c}}}}{{\text{a}}}} \right) \\

\Rightarrow {\text{Sin}}\left( {\dfrac{{{\text{B - C}}}}{2}} \right) = \left( {\dfrac{{{\text{b - c}}}}{2}} \right){\text{Cos}}\dfrac{{\text{A}}}{2} \\

$

Hence proved.

**Note:**In order to solve these types of questions the key is to just simplify the given using suitable trigonometric formulas. Drawing diagrams will make the solution a bit easier. Here we have used lami’s theorem. Identifying that the sum of all angles here is 180° is the key step which helps us solve the problem. Good knowledge in trigonometric formulae is required.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

At which age domestication of animals started A Neolithic class 11 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE