Advertisements
Advertisements
Question
In triangle ABC, prove the following:
Advertisements
Solution
Let
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]
Then,
Consider the LHS of the equation
\[LHS = \frac{a^2 \sin\left( B - C \right)}{\sin A} + \frac{b^2 \sin\left( C - A \right)}{\sin B} + \frac{c^2 \sin\left( A - B \right)}{\sin C}\]
\[ = \frac{k^2 \sin^2 A\sin\left( B - C \right)}{\sin A} + \frac{k^2 \sin^2 B\sin\left( C - A \right)}{\sin B} + \frac{k^2 \sin^2 C\sin\left( A - B \right)}{\sin C} \]
\[ = k^2 \sin A\sin\left( B - C \right) + k^2 \sin B\sin\left( C - A \right) + k^2 \sin C\sin\left( A - B \right) \]
\[ = k^2 \left[ \sin A\left( \sin B\cos C - \sin C\cos B \right) + \sin B\left( \sin C\cos A - \sin A\cos C \right) + \sin C\left( \sin A\cos B - \sin B\cos A \right) \right] \]
\[ = k^2 \left( \sin A\sin B\cos C - \sin A\sin C\cos B + \sin B\sin C\cos A - \sin A\sin B\cos C + \sin A\sin C\cos B - \sin C\sin B\cos A \right)\]
\[ = 0 = RHS\]
\[\text{ Hence proved } .\]
APPEARS IN
RELATED QUESTIONS
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In any triangle ABC, prove the following:
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 ∆ABC, prove that: \[a \sin\frac{A}{2} \sin \left( \frac{B - C}{2} \right) + b \sin \frac{B}{2} \sin \left( \frac{C - A}{2} \right) + c \sin \frac{C}{2} \sin \left( \frac{A - B}{2} \right) = 0\]
In ∆ABC, prove that: \[\frac{b \sec B + c \sec C}{\tan B + \tan C} = \frac{c \sec C + a \sec A}{\tan C + \tan A} = \frac{a \sec A + b \sec B}{\tan A + \tan B}\]
In triangle ABC, prove the following:
In ∆ABC, prove that if θ be any angle, then b cosθ = c cos (A − θ) + a cos (C + θ).
In ∆ABC, if sin2 A + sin2 B = sin2 C. show that the triangle is right-angled.
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.
A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c metres along a slope inclined at an angle β and finds the angle of elevation of the peak of the hill to be ϒ. Show that the height of the peak above the ground is \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]
In \[∆ ABC, if a = \sqrt{2}, b = \sqrt{3} \text{ and } c = \sqrt{5}\] show that its area is \[\frac{1}{2}\sqrt{6} sq .\] units.
In ∆ ABC, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C.
In ∆ABC, prove the following: \[c \left( a \cos B - b \cos A \right) = a^2 - b^2\]
In ∆ABC, prove the following:
\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\]
In ∆ABC, prove the following:
\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]
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, prove the following:
\[\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C\]
In \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\] Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
Two ships leave a port at the same time. One goes 24 km/hr in the direction N 38° E and other travels 32 km/hr in the direction S 52° E. Find the distance between the ships at the end of 3 hrs.
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 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 \[\angle C\]
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.
Mark the correct alternative in each of the following:
In any ∆ABC, \[\sum^{}_{} a^2 \left( \sin B - \sin C \right)\] =
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 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 = 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)]
