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.
In triangle ABC, prove the following:
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:
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In ∆ABC, prove that: \[\frac{b \sec B + c \sec C}{\tan B + \tan C} = \frac{c \sec C + a \sec A}{\tan C + \tan A} = \frac{a \sec A + b \sec B}{\tan A + \tan B}\]
\[a \left( \cos B \cos C + \cos A \right) = b \left( \cos C \cos A + \cos B \right) = c \left( \cos A \cos B + \cos C \right)\]
In ∆ABC, prove that \[a \left( \cos C - \cos B \right) = 2 \left( b - c \right) \cos^2 \frac{A}{2} .\]
In ∆ABC, if sin2 A + sin2 B = sin2 C. show that the triangle is right-angled.
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.
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 = 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:
\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]
In ∆ABC, prove the following:
\[4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2\]
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.
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 =\[\sqrt{3}\] 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 ∆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:
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 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, 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)`
If x = sec Φ – tan Φ and y = cosec Φ + cot Φ then show that xy + x – y + 1 = 0
[Hint: Find xy + 1 and then show that x – y = –(xy + 1)]
