Advertisements
Advertisements
Question
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
\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]
Advertisements
Solution
Using sine rule, we have
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]
\[ \Rightarrow a = k\sin A, b = k\sin B, c = k\sin C\]
\[\therefore \sum^{}_{}a\left( \sin B - \sin C \right)\]
\[ = \sum^{}_{}k\sin A\left( \sin B - \sin C \right)\]
\[= k\sum^{}_{}\sin A\left( \sin B - \sin C \right)\]
\[= k\left[ \sin A\left( \sin B - \sin C \right) + \sin B\left( \sin C - \sin A \right) + \sin C\left( \sin A - \sin B \right) \right]\] \[= k\left( \sin A\sin B - \sin A\sin C + \sin B\sin C - \sin B\sin A + \sin C\sin A - \sin C\sin B \right)\]
\[= k\times0=0\]
APPEARS IN
RELATED QUESTIONS
If in ∆ABC, ∠A = 45°, ∠B = 60° and ∠C = 75°, find the ratio of its sides.
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 triangle ABC, prove the following:
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 a2, b2 and c2 are in A.P., prove that cot A, cot B and cot C are also in A.P.
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.
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, if a = 18, b = 24 and c = 30, find cos A, cos B and cos C.
In ∆ABC, prove the following: \[b \left( c \cos A - a \cos C \right) = c^2 - a^2\]
In ∆ABC, prove the following:
\[2 \left( bc \cos A + ca \cos B + ab \cos C \right) = a^2 + b^2 + c^2\]
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 \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\] Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
In \[∆ ABC, if \angle B = 60°,\] prove that \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]
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 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 ∆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.
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)\
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 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)`
