Advertisements
Advertisements
Question
In ∆ABC, prove that: \[a \sin\frac{A}{2} \sin \left( \frac{B - C}{2} \right) + b \sin \frac{B}{2} \sin \left( \frac{C - A}{2} \right) + c \sin \frac{C}{2} \sin \left( \frac{A - B}{2} \right) = 0\]
Solution
Consider
\[a\sin\frac{A}{2}\sin\left( \frac{B - C}{2} \right) + b\sin\frac{B}{2}\sin\left( \frac{C - A}{2} \right) + c\sin\frac{C}{2}\sin\left( \frac{A - B}{2} \right)\]
\[ = k\left[ \sin\left\{ \pi - \left( B + C \right) \right\}\sin\frac{A}{2}\sin\left( \frac{B - C}{2} \right) + \sin\left\{ \pi - \left( C + A \right) \right\} \sin\frac{B}{2}\sin\left( \frac{C - A}{2} \right) + \sin\left\{ \pi - \left( A + B \right) \right\}\sin\frac{C}{2}\sin\left( \frac{A - B}{2} \right) \right] \left( \because A + B + C = \pi \right)\]
\[ = k\left[ \sin\left( B + C \right)\sin\frac{A}{2}\sin\left( \frac{B - C}{2} \right) + \sin\left( A + C \right)\sin\frac{B}{2}\sin\left( \frac{C - A}{2} \right) + \sin\left( A + B \right)\sin\frac{C}{2}\sin\left( \frac{A - B}{2} \right) \right]\]
\[ = k\left[ 2\sin\left( \frac{B + C}{2} \right)\cos\left( \frac{B - C}{2} \right)\sin\frac{A}{2}\sin\left( \frac{B - C}{2} \right) + 2\sin\left( \frac{A + C}{2} \right)\cos\left( \frac{C - A}{2} \right)\sin\frac{B}{2}\sin\left( \frac{C - A}{2} \right) + 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)\sin\frac{C}{2}\sin\left( \frac{A - B}{2} \right) \right]\]
\[ = 2k\left[ \sin\left( \frac{B + C}{2} \right)\sin\frac{A}{2}\sin\frac{A}{2}\sin\left( \frac{B - C}{2} \right) + \sin\left( \frac{A + C}{2} \right)\sin\frac{B}{2}\sin\frac{B}{2}\sin\left( \frac{C - A}{2} \right) + \sin\left( \frac{A + B}{2} \right)\sin\frac{C}{2}\sin\frac{C}{2}\sin\left( \frac{A - B}{2} \right) \right]\]
\[ = 2k\left[ \sin\left( \frac{B + C}{2} \right)\sin\left( \frac{B - C}{2} \right) \sin^2 \frac{A}{2} + \sin\left( \frac{A + C}{2} \right)\sin\left( \frac{C - A}{2} \right) \sin^2 \frac{B}{2} + \sin\left( \frac{A + B}{2} \right)\sin\left( \frac{A - B}{2} \right) \sin^2 \frac{C}{2} \right]\]
\[ = 2k \sin^2 \frac{A}{2}\left( \sin^2 \frac{B}{2} - \sin^2 \frac{C}{2} \right) + 2k \sin^2 \frac{B}{2}\left( \sin^2 \frac{C}{2} - \sin^2 \frac{A}{2} \right) + 2k \sin^2 \frac{C}{2}\left( \sin^2 \frac{A}{2} - \sin^2 \frac{B}{2} \right)\]
\[ = 2k\left( \sin^2 \frac{A}{2} \sin^2 \frac{B}{2} - \sin^2 \frac{A}{2} \sin^2 \frac{C}{2} + \sin^2 \frac{B}{2} \sin^2 \frac{C}{2} - \sin^2 \frac{A}{2} \sin^2 \frac{B}{2} + \sin^2 \frac{A}{2} \sin^2 \frac{C}{2} - \sin^2 \frac{C}{2} \sin^2 \frac{B}{2} \right)\]
\[ = k\left( 0 \right)\]
\[ = 0\]
Hence proved.
APPEARS IN
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:
\[\left( a - b \right) \cos \frac{C}{2} = c \sin \left( \frac{A - B}{2} \right)\]
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:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\]
In ∆ABC, if a2, b2 and c2 are in A.P., prove that cot A, cot B and cot C are also in A.P.
The upper part of a tree broken by the wind makes an angle of 30° with the ground and the distance from the root to the point where the top of the tree touches the ground is 15 m. Using sine rule, find the height of the tree.
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 a, b and c of ∆ABC are in H.P., prove that \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]
In \[∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°\] show that its area is \[\frac{15\sqrt{3}}{2} sq\].units.
In ∆ ABC, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C.
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 \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\] Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
Answer the following questions in one word or one sentence or as per exact requirement of the question.
Find the area of the triangle ∆ABC in which a = 1, b = 2 and \[\angle C = 60º\]
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In a ∆ABC, if b = 20, c = 21 and \[\sin A = \frac{3}{5}\]
Answer the following questions in one word or one sentence or as per exact requirement of the question.
In a ∆ABC, if sinA and sinB are the roots of the equation \[c^2 x^2 - c\left( a + b \right)x + ab = 0\] then find \[\angle C\]
Mark the correct alternative in each of the following:
In a ∆ABC, if a = 2, \[\angle B = 60°\] and\[\angle C = 75°\]
Mark the correct alternative in each of the following:
If the sides of a triangle are in the ratio \[1: \sqrt{3}: 2\] then the measure of its greatest angle is
Mark the correct alternative in each of the following:
In any ∆ABC, 2(bc cosA + ca cosB + ab cosC) =
Mark the correct alternative in each of the following:
In any ∆ABC, the value of \[2ac\sin\left( \frac{A - B + C}{2} \right)\] is
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)]
