Advertisements
Advertisements
प्रश्न
If in \[∆ ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1\] prove that the triangle is right-angled.
Advertisements
उत्तर
Let ABC be any triangle.
In \[∆ ABC\]
\[\cos^2 A + \cos^2 B + \cos^2 C = 1\]
\[ \Rightarrow \cos^2 A + \cos^2 B + \cos^2 \left[ \pi - \left( B + A \right) \right] = 1 \left( \because A + B + C = \pi \right)\]
\[ \Rightarrow \cos^2 A + \cos^2 B + \cos^2 \left( B + A \right) = 1\]
\[ \Rightarrow \cos^2 A + \cos^2 B = 1 - \cos^2 \left( B + A \right)\]
\[ \Rightarrow \cos^2 A + \cos^2 B = \sin^2 \left( B + A \right)\]
\[ \Rightarrow \cos^2 A + \cos^2 B = \left( \sin A\cos B + \cos A\sin B \right)^2 \]
\[ \Rightarrow \cos^2 A + \cos^2 B = \sin^2 A \cos^2 B + \cos^2 A \sin^2 B + 2\sin A\sin B\cos A\cos B\]
\[ \Rightarrow \cos^2 A\left( 1 - \sin^2 B \right) + \cos^2 B\left( 1 - \sin^2 A \right) = 2\sin A\sin B\cos A\cos B\]
\[ \Rightarrow 2 \cos^2 A \cos^2 B = 2\sin A\sin B\cos A\cos B\]
\[ \Rightarrow \cos A\cos B = \sin A\sin B\]
\[ \Rightarrow \cos A\cos B - \sin A\sin B = 0\]
\[ \Rightarrow \cos \left( A + B \right) = 0\]
\[ \Rightarrow \cos \left( A + B \right) = \cos {90}^° \]
\[ \Rightarrow A + B = {90}^°\]
\[ \Rightarrow C = {90}^° \left( \because A + B + C = {180}^° \right)\]
Hence, \[∆\]ABC is right angled.
APPEARS IN
संबंधित प्रश्न
If in ∆ABC, ∠C = 105°, ∠B = 45° and a = 2, then find b.
In ∆ABC, if a = 18, b = 24 and c = 30 and ∠c = 90°, find sin A, sin B and sin C.
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 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, 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.
In \[∆ ABC, if a = 5, b = 6 a\text{ and } C = 60°\] show that its area is \[\frac{15\sqrt{3}}{2} sq\].units.
In \[∆ ABC, if a = \sqrt{2}, b = \sqrt{3} \text{ and } c = \sqrt{5}\] show that its area is \[\frac{1}{2}\sqrt{6} sq .\] 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:
\[\frac{c - b \cos A}{b - c \cos A} = \frac{\cos B}{\cos C}\]
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, prove the following:
\[\sin^3 A \cos \left( B - C \right) + \sin^3 B \cos \left( C - A \right) + \sin^3 C \cos \left( A - B \right) = 3 \sin A \sin B \sin C\]
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.
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 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)]
