Advertisements
Advertisements
प्रश्न
In ∆ABC, prove that if θ be any angle, then b cosθ = c cos (A − θ) + a cos (C + θ).
उत्तर
Suppose
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\] ...(1)
Consider the RHS of the equation b cosθ = c cos (A − θ) + a cos (C + θ).
\[RHS = c\cos\left( A - \theta \right) + a\cos\left( C + \theta \right)\]
\[ = k\sin C\cos\left( A - \theta \right) + k\sin A\cos\left( C + \theta \right) \left( from \left( 1 \right) \right) \]
\[ = \frac{k}{2}\left[ 2\sin C\cos\left( A - \theta \right) + 2\sin A\cos\left( C + \theta \right) \right]\]
\[ = \frac{k}{2}\left[ \sin\left( A + C - \theta \right) + \sin\left( C + \theta - A \right) + \sin\left( A + C + \theta \right) + \sin\left( A - C - \theta \right) \right] \]
\[ = \frac{k}{2}\left( \sin\left( \pi - B - \theta \right) + \sin\left( C + \theta - A \right) + \sin\left( \pi - B + \theta \right) - \sin\left( C + \theta - A \right) \right) \left( \because A + B + C = \pi \right)\]
\[ = \frac{k}{2}\left( \sin\left( B + \theta \right) + \sin\left( B - \theta \right) \right)\]
\[ = \frac{k}{2}\left( \sin B\cos\theta + \sin\theta\cos B + \sin B\cos\theta - \sin\theta\cos B \right)\]
\[ = \frac{k}{2}\left( 2\sin B\cos\theta \right)\]
\[ = k\sin B\cos\theta\]
\[ = b\cos\theta = LHS\]
\[\text{ Hence proved } .\]
APPEARS IN
संबंधित प्रश्न
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:
\[\left( a - b \right) \cos \frac{C}{2} = c \sin \left( \frac{A - B}{2} \right)\]
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 ∆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 ∆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.
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:
\[a^2 = \left( b + c \right)^2 - 4 bc \cos^2 \frac{A}{2}\]
In ∆ABC, prove the following:
\[4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2\]
In ∆ABC, prove the following:
\[\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C\]
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.
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 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, \[a\left( b\cos C - c\cos B \right) =\]
Find the value of `(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8)`
