Advertisements
Advertisements
Question
a cos A + b cos B + c cos C = 2b sin A sin C
Solution
\[\text{ By sine rule, we know that }\]
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \left( say \right)\]
\[ \Rightarrow a = k \sin A, b = k \sin B, c = k \sin C\]
\[\text{ Now }, \]
\[LHS = a \cos A + b \cos B + c \cos C\]
\[ = k \sin A \cos A + k \sin B \cos B + k \sin C \cos C\]
\[ = \frac{k}{2} \left( 2 \sin A \cos A + 2 \sin B \cos B + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( \sin 2A + \sin 2B + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \frac{2A + 2B}{2}\cos\frac{2A - 2B}{2} + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \left( A + B \right) \cos \left( A - B \right) + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \left( 2 \sin \left( \pi - C \right) \cos \left( A - B \right) + 2 \sin C \cos C \right) \left( \because A + B + C = \pi \right)\]
\[ = \frac{k}{2} \left( 2 \sin C \cos \left( A - B \right) + 2 \sin C \cos C \right)\]
\[ = \frac{k}{2} \times 2 \sin C\left( \cos \left( A - B \right) + \cos C \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{A - B + C}{2} \right)\cos \left( \frac{A - B - C}{2} \right) \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi - B - B}{2} \right)\cos \left( \frac{B + C - A}{2} \right) \right) \left( \because A + B + C = \pi \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi - 2B}{2} \right)\cos \left( \frac{\pi - 2A}{2} \right) \right) \left( \because A + B + C = \pi \right)\]
\[ = k \sin C\left( 2 \cos \left( \frac{\pi}{2} - B \right)\cos \left( \frac{\pi}{2} - A \right) \right)\]
\[ = 2k \sin C\left( \sin B \sin A \right)\]
\[ = 2 \left( k \sin B \right) \sin A \sin C\]
\[ = 2b \sin A \sin C\]
\[ = RHS\]
\[ \therefore LHS = RHS\]
Hence, a cos A + b cos B + c cos C = 2b sin A sin C.
APPEARS IN
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 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 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:
\[a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)\]
In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\]
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.
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, 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 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:
\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]
In \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\] Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
If in \[∆ ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1\] prove that the triangle is right-angled.
In \[∆ ABC \text{ if } \cos C = \frac{\sin A}{2 \sin B}\] prove that the triangle is isosceles.
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 b =\[\sqrt{3}\] and \[\angle A = 30°\] find 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.
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:
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, \[a\left( b\cos C - c\cos B \right) =\]
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.
