Advertisements
Advertisements
प्रश्न
In triangle ABC, prove the following:
Advertisements
उत्तर
Let
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\] ...(1)
We need to prove:
\[ = \frac{ksinC}{k\left( sinA + sinB \right)} \left( using\left( 1 \right) \right)\]
\[ = \frac{2\sin\frac{C}{2}\cos\frac{C}{2}}{2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}\]
\[ = \frac{\sin\frac{C}{2}\cos\left( \frac{\pi - \left( A + B \right)}{2} \right)}{\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)} \left( \because A + B + C = \pi \right)\]
\[ = \frac{\sin\frac{C}{2}\sin\left( \frac{A + B}{2} \right)}{\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}\]
\[ = \frac{\sin\frac{C}{2}}{\cos\left( \frac{A - B}{2} \right)} . . . \left( 2 \right)\]
\[RHS = \frac{1 - tan\frac{A}{2}tan\frac{B}{2}}{1 + tan\frac{A}{2}tan\frac{B}{2}}\]
\[ = \frac{1 - \frac{sin\frac{A}{2}}{cos\frac{A}{2}}\frac{sin\frac{B}{2}}{cos\frac{B}{2}}}{1 + \frac{sin\frac{A}{2}}{cos\frac{A}{2}}\frac{sin\frac{B}{2}}{cos\frac{B}{2}}}\]
\[ = \frac{cos\frac{A}{2}cos\frac{B}{2} - sin\frac{A}{2}sin\frac{B}{2}}{cos\frac{A}{2}cos\frac{B}{2} + sin\frac{A}{2}sin\frac{B}{2}}\]
\[ = \frac{cos\left( \frac{A + B}{2} \right)}{cos\left( \frac{A - B}{2} \right)}\]
\[ = \frac{cos\left( \frac{\pi - C}{2} \right)}{cos\left( \frac{A - B}{2} \right)} \left( \because A + B + C = \pi \right)\]
\[ = \frac{\sin\frac{C}{2}}{\cos\left( \frac{A - B}{2} \right)} = LHS \left( \text{ from }\left( 2 \right) \right)\]
\[\text{ Hence proved } .\]
APPEARS IN
संबंधित प्रश्न
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 triangle ABC, prove the following:
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: \[b \left( c \cos A - a \cos C \right) = c^2 - a^2\]
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 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.
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 \[\cos A = \frac{\sin B}{2\sin C}\] then show that c = 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.
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 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
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)]
