Advertisements
Advertisements
प्रश्न
In triangle ABC, prove the following:
उत्तर
Let
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\]
Then,
Consider the LHS of the equation
\[ = \frac{1 - 2 \sin^2 A}{a^2} - \frac{1 - 2 \sin^2 B}{b^2} \]
\[ = \frac{1 - 2\frac{a^2}{k^2}}{a^2} - \frac{1 - 2\frac{b^2}{k^2}}{b^2} \]
\[ = \frac{\frac{k^2 - 2 a^2}{k^2}}{a^2} - \frac{\frac{k^2 - 2 b^2}{k^2}}{b^2}\]
\[ = \frac{k^2 b^2 - 2 a^2 b^2 - k^2 a^2 + 2 a^2 b^2}{a^2 b^2}\]
\[ = \frac{k^2 \left( b^2 - a^2 \right)}{k^2 a^2 b^2}\]
\[ = \frac{1}{a^2} - \frac{1}{b^2} = RHS\]
\[\text { Hence proved } .\]
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 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:
\[\frac{a^2 - c^2}{b^2} = \frac{\sin \left( A - C \right)}{\sin \left( A + C \right)}\]
In triangle ABC, prove the following:
In triangle ABC, prove the following:
In ∆ABC, prove that if θ be any angle, then b cosθ = c cos (A − θ) + a cos (C + θ).
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.
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 \[\frac{c \sin \alpha \sin \left( \gamma - \beta \right)}{\left( \sin \gamma - \alpha \right)}\]
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, 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:
\[\left( c^2 - a^2 + b^2 \right) \tan A = \left( a^2 - b^2 + c^2 \right) \tan B = \left( b^2 - c^2 + a^2 \right) \tan C\]
In ∆ABC, prove the following:
\[\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C\]
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 \[\cos A = \frac{\sin B}{2\sin C}\] then show that c = a.
Answer the following questions in one word or one sentence or as per exact requirement of the question.
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.
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:
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 any ∆ABC, the value of \[2ac\sin\left( \frac{A - B + C}{2} \right)\] 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)`
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)]
