Advertisements
Advertisements
प्रश्न
a cos A + b cos B + c cos C = 2b sin A sin C
उत्तर
\[\text{ By sine rule, we know that }\]
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \left( say \right)\]
\[ \Rightarrow a = k \sin A, b = k \sin B, c = k \sin C\]
\[\text{ Now }, \]
\[LHS = a \cos A + b \cos B + c \cos C\]
\[ = k \sin A \cos A + k \sin B \cos B + k \sin C \cos C\]
\[ = \frac{k}{2} \left( 2 \sin A \cos A + 2 \sin B \cos B + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( \sin 2A + \sin 2B + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \frac{2A + 2B}{2}\cos\frac{2A - 2B}{2} + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \left( A + B \right) \cos \left( A - B \right) + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \left( \pi - C \right) \cos \left( A - B \right) + 2 \sin C \cos C \right) \left( \because A + B + C = \pi \right)\]
\[ = \frac{k}{2} \left( 2 \sin C \cos \left( A - B \right) + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \times 2 \sin C\left( \cos \left( A - B \right) + \cos C \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{A - B + C}{2} \right)\cos \left( \frac{A - B - C}{2} \right) \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi - B - B}{2} \right)\cos \left( \frac{B + C - A}{2} \right) \right) \left( \because A + B + C = \pi \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi - 2B}{2} \right)\cos \left( \frac{\pi - 2A}{2} \right) \right) \left( \because A + B + C = \pi \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi}{2} - B \right)\cos \left( \frac{\pi}{2} - A \right) \right)\]
\[ = 2k \sin C\left( \sin B \sin A \right)\]
\[ = 2 \left( k \sin B \right) \sin A \sin C\]
\[ = 2b \sin A \sin C\]
\[ = RHS\]
\[ \therefore LHS = RHS\]
Hence, a cos A + b cos B + c cos C = 2b sin A sin C.
APPEARS IN
संबंधित प्रश्न
If in ∆ABC, ∠A = 45°, ∠B = 60° and ∠C = 75°, find the ratio of its sides.
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:
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: \[\frac{b \sec B + c \sec C}{\tan B + \tan C} = \frac{c \sec C + a \sec A}{\tan C + \tan A} = \frac{a \sec A + b \sec B}{\tan A + \tan B}\]
In ∆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.
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.
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 \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]
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 \[∆ ABC, if a = \sqrt{2}, b = \sqrt{3} \text{ and } c = \sqrt{5}\] show that its area is \[\frac{1}{2}\sqrt{6} sq .\] units.
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:
\[\left( c^2 - a^2 + b^2 \right) \tan A = \left( a^2 - b^2 + c^2 \right) \tan B = \left( b^2 - c^2 + a^2 \right) \tan C\]
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 \[∆ ABC, if \angle B = 60°,\] prove that \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]
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 \[\angle C\]
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 a ∆ABC, if a = 2, \[\angle B = 60°\] and\[\angle C = 75°\]
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, \[a\left( b\cos C - c\cos B \right) =\]
