Advertisements
Advertisements
Question
In triangle ABC, prove the following:
Solution
\[Let\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k . . . \left( 1 \right)\]
\[\text{ Consider the LHS of the equation a }\cos A + b\cos B + c \cos C . \]
\[a\cos A + b\cos B + c\cos C = k\left( \sin A\cos A + \sin B\cos B + \sin C\cos C \right) \]
\[ = \frac{k}{2}\left( 2sinAcosA + 2sinAcosA + 2sinCcosC \right)\]
\[ = \frac{k}{2}\left( \sin2A + \sin2 B + \sin2 C \right)\]
\[ = \frac{k}{2}\left[ 2\sin\left( A + B \right)\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{k}{2}\left[ 2\sin\left( \pi - C \right)\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{k}{2}\left[ 2\sin C\cos\left( A - B \right) + 2\sin C\cos C \right]\]
\[ = \frac{2k\sin C}{2}\left[ \cos\left( A - B \right) + \cos C \right] \]
\[= k\sin C\left[ \cos\left( A - B \right) + \cos\left\{ \pi - \left( A + B \right) \right\} \right]\]
\[ = k\sin C\left[ \cos\left( A - B \right) - \cos\left( A + B \right) \right]\]
\[ = k\sin C\left[ 2\sin Asin B \right]\]
\[ = 2k\sin A\sin B\sin C . . . (1)\]
\[\text{ Now }, \]
\[\text{ on putting } k\sin C = \text{ C in equation } (1), \text{ we get }: \]
\[2c\sin A\sin B\]
\[\text{ and on putting k }\sin B = \text{ b in equation } (1), \text{ we get }: \]
\[2b\sin A\sin C\]
So, from (1), we have
Hence proved.
APPEARS IN
RELATED QUESTIONS
If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b.
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.
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 \[\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2} \text{ and } \sin^2 \frac{C}{2}\]
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:
\[2 \left( bc \cos A + ca \cos B + ab \cos C \right) = a^2 + b^2 + c^2\]
In ∆ABC, prove the following:
\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\]
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 \[\left( a + b + c \right) \left( a - b + c \right) = 3ca\]
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.
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 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 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 any ∆ABC, 2(bc cosA + ca cosB + ab cosC) =
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 a ∆ABC, if \[\left( c + a + b \right)\left( a + b - c \right) = ab\] then the measure of angle C is
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 cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`, then find the value of xy + yz + zx.
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)]
