English

In Triangle Abc, Prove the Following: a 2 Sin ( B − C ) = ( B 2 − C 2 ) Sin a

Advertisements
Advertisements

Question

In triangle ABC, prove the following: 

\[a^2 \sin \left( B - C \right) = \left( b^2 - c^2 \right) \sin A\]

 

Advertisements

Solution

Let 

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\] 

Consider the RHS of the equation

\[a^2 \sin \left( B - C \right) = \left( b^2 - c^2 \right) \sin A\]

\[RHS = k^2 \sin A\left( \sin^2 B - \sin^2 C \right) \]
\[ = k^2 \sin A\left[ \sin\left( B + C \right)\sin\left( B - C \right) \right] \left[ \because \sin^2 B - \sin^2 C = \sin\left( B + C \right)\sin\left( B - C \right) \right]\]
\[ = k^2 \sin A\left[ \sin\left( \pi - A \right)\sin\left( B - C \right) \right] \left[ \because A + B + C = \pi \right]\]
\[ = k^2 \sin A\left[ \sin\left( A \right)\sin\left( B - C \right) \right]\]
\[ = k^2 \sin^2 A\sin\left( B - C \right)\]
\[ = a^2 \sin\left( B - C \right) = LHS \left[ \because a = k\sin A \right]\]
\[\text{ Hence proved } .\]

shaalaa.com
Sine and Cosine Formulae and Their Applications
  Is there an error in this question or solution?
Chapter 10: Sine and cosine formulae and their applications - Exercise 10.1 [Page 13]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 10 Sine and cosine formulae and their applications
Exercise 10.1 | Q 12 | Page 13

RELATED QUESTIONS

In ∆ABC, if a = 18, b = 24 and c = 30 and ∠c = 90°, find sin A, sin B and sin C


In triangle ABC, prove the following: 

\[\frac{a - b}{a + b} = \frac{\tan \left( \frac{A - B}{2} \right)}{\tan \left( \frac{A + B}{2} \right)}\]

 


In triangle ABC, prove the following:

\[\frac{c}{a - b} = \frac{\tan\left( \frac{A}{2} \right) + \tan \left( \frac{B}{2} \right)}{\tan \left( \frac{A}{2} \right) - \tan \left( \frac{B}{2} \right)}\]

 


In triangle ABC, prove the following: 

\[\frac{a + b}{c} = \frac{\cos \left( \frac{A - B}{2} \right)}{\sin \frac{C}{2}}\]

 


In triangle ABC, prove the following: 

\[\frac{a^2 - c^2}{b^2} = \frac{\sin \left( A - C \right)}{\sin \left( A + C \right)}\] 


In triangle ABC, prove the following: 

\[\frac{\sqrt{\sin A} - \sqrt{\sin B}}{\sqrt{\sin A} + \sqrt{\sin B}} = \frac{a + b - 2\sqrt{ab}}{a - b}\]

 


In triangle ABC, prove the following: 

\[a \left( \sin B - \sin C \right) + \left( \sin C - \sin A \right) + c \left( \sin A - \sin B \right) = 0\]

 


In triangle ABC, prove the following: 

\[\frac{a^2 \sin \left( B - C \right)}{\sin A} + \frac{b^2 \sin \left( C - A \right)}{\sin B} + \frac{c^2 \sin \left( A - B \right)}{\sin C} = 0\]

 


In triangle ABC, prove the following: 

\[b \cos B + c \cos C = a \cos \left( B - C \right)\]

 


In triangle ABC, prove the following:

\[\frac{\cos 2A}{a^2} - \frac{\cos 2B}{b^2} - \frac{1}{a^2} - \frac{1}{b^2}\]

 


In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\] 


At the foot of a mountain, the elevation of it summit is 45°; after ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain. 


If the sides ab and c of ∆ABC are in H.P., prove that \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]


The sides of a triangle are a = 4, b = 6 and c = 8. Show that \[8 \cos A + 16 \cos B + 4 \cos C = 17\]


In ∆ABC, prove the following: \[b \left( c \cos A - a \cos C \right) = c^2 - a^2\]


In ∆ABC, prove that  \[a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0\]


In ∆ABC, prove the following:

\[4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2\]


In \[∆ ABC \text{ if } \cos C = \frac{\sin A}{2 \sin B}\] prove that the triangle is isosceles.  


Answer  the following questions in one word or one sentence or as per exact requirement of the question. 

In a ∆ABC, if \[\cos A = \frac{\sin B}{2\sin C}\]  then show that c = a


Answer the following questions in one word or one sentence or as per exact requirement of the question.  

In ∆ABC, if a = 8, b = 10, c = 12 and C = λA, find the value of λ


Answer the following questions in one word or one sentence or as per exact requirement of the question. 

If the sides of a triangle are proportional to 2, \[\sqrt{6}\] and \[\sqrt{3} - 1\] find the measure of its greatest angle. 


Answer the following questions in one word or one sentence or as per exact requirement of the question. 

In any triangle ABC, find the value of \[a\sin\left( B - C \right) + b\sin\left( C - A \right) + c\sin\left( A - B \right)\ 


Answer the following questions in one word or one sentence or as per exact requirement of the question. 

In any ∆ABC, find the value of

\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]


Mark the correct alternative in each of the following: 

In a triangle ABC, a = 4, b = 3, \[\angle A = 60°\]   then c is a root of the equation 


Mark the correct alternative in each of the following: 

In a ∆ABC, if  \[\left( c + a + b \right)\left( a + b - c \right) = ab\] then the measure of angle C is 


Mark the correct alternative in each of the following:

In any ∆ABC, \[a\left( b\cos C - c\cos B \right) =\]  


Find the value of `(1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8)`


If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`, then find the value of xy + yz + zx.


If x = sec Φ – tan Φ and y = cosec Φ + cot Φ then show that xy + x – y + 1 = 0
[Hint: Find xy + 1 and then show that x – y = –(xy + 1)]


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×