Advertisements
Advertisements
प्रश्न
In triangle ABC, prove the following:
Advertisements
उत्तर
\[Let\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k . . . \left( 1 \right)\]
\[\text{ Consider the LHS of the equation a }\cos A + b\cos B + c \cos C . \]
\[a\cos A + b\cos B + c\cos C = k\left( \sin A\cos A + \sin B\cos B + \sin C\cos C \right) \]
\[ = \frac{k}{2}\left( 2sinAcosA + 2sinAcosA + 2sinCcosC \right)\]
\[ = \frac{k}{2}\left( \sin2A + \sin2 B + \sin2 C \right)\]
\[ = \frac{k}{2}\left[ 2\sin\left( A + B \right)\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{k}{2}\left[ 2\sin\left( \pi - C \right)\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{k}{2}\left[ 2\sin C\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{2k\sin C}{2}\left[ \cos\left( A - B \right) + \cos C \right] \]
\[= k\sin C\left[ \cos\left( A - B \right) + \cos\left\{ \pi - \left( A + B \right) \right\} \right]\]
\[ = k\sin C\left[ \cos\left( A - B \right) - \cos\left( A + B \right) \right]\]
\[ = k\sin C\left[ 2\sin Asin B \right]\]
\[ = 2k\sin A\sin B\sin C . . . (1)\]
\[\text{ Now }, \]
\[\text{ on putting } k\sin C = \text{ C in equation } (1), \text{ we get }: \]
\[2c\sin A\sin B\]
\[\text{ and on putting k }\sin B = \text{ b in equation } (1), \text{ we get }: \]
\[2b\sin A\sin C\]
So, from (1), we have
Hence proved.
APPEARS IN
संबंधित प्रश्न
If in ∆ABC, ∠A = 45°, ∠B = 60° and ∠C = 75°, find the ratio of its sides.
If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b.
In ∆ABC, if a = 18, b = 24 and c = 30 and ∠c = 90°, find sin A, sin B and sin C.
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 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 \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\]
In ∆ABC, if sin2 A + sin2 B = sin2 C. show that the triangle is right-angled.
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.
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.
If the sides a, b and c of ∆ABC are in H.P., prove that \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]
a cos A + b cos B + c cos C = 2b sin A sin C
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 \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]
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.
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.
Answer the following questions in one word or one sentence or as per exact requirement of the question.
If in a ∆ABC, \[\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}\] then find the measures of angles A, B, C.
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
\[\sum^{}_{}a\left( \text{ sin }B - \text{ sin }C \right)\]
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 a = 2, \[\angle B = 60°\] and\[\angle C = 75°\]
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, the value of \[2ac\sin\left( \frac{A - B + C}{2} \right)\] is
If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`, then find the value of xy + yz + zx.
