Advertisements
Advertisements
प्रश्न
a cos A + b cos B + c cos C = 2b sin A sin C
Advertisements
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.
संबंधित प्रश्न
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:
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: \[a \sin\frac{A}{2} \sin \left( \frac{B - C}{2} \right) + b \sin \frac{B}{2} \sin \left( \frac{C - A}{2} \right) + c \sin \frac{C}{2} \sin \left( \frac{A - B}{2} \right) = 0\]
In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\]
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 \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]
In \[∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°\] show that its area is \[\frac{15\sqrt{3}}{2} sq\].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: \[c \left( a \cos B - b \cos A \right) = a^2 - b^2\]
In ∆ABC, prove the following:
\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\]
In ∆ABC, prove 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 \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
In \[∆ ABC, if \angle B = 60°,\] prove that \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]
If in \[∆ ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1\] prove that the triangle is right-angled.
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 = 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.
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 ∆ABC, if \[\left( c + a + b \right)\left( a + b - c \right) = ab\] then the measure of angle C is
