Advertisements
Advertisements
Question
In ∆ABC, prove that \[a \left( \cos B + \cos C - 1 \right) + b \left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right) = 0\]
Advertisements
Solution
\[\text{ Consider the LHS of the given equation } . \]
\[LHS = a\left( \cos B + \cos C - 1 \right) + b\left( \cos C + \cos A - 1 \right) + c\left( \cos A + \cos B - 1 \right)\]
\[ = a\cos B + b\cos C + a\cos C + b\cos A + c\cos A + c\cos B - \left( a + b + c \right)\]
\[ = \left( a\cos B + b\cos A \right) + \left( b\cos C + c\cos B \right) + \left( a\cos C + c\cos A \right) - \left( a + b + c \right) \]
\[ = c + a + b - \left( a + b + c \right) . . \left( \text{ Using projection formula }: a = b\cos C + c\cos B, b = a\cos C + c\cos A, c = a\cos B + b\cos A \right)\]
\[ = 0 = RHS \]
\[\text{ Hence proved }.\]
APPEARS IN
RELATED QUESTIONS
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:
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 ∆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 ∆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 a2, b2 and c2 are in A.P., prove that cot A, cot B and cot C are also in A.P.
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:
\[4\left( bc \cos^2 \frac{A}{2} + ca \cos^2 \frac{B}{2} + ab \cos^2 \frac{C}{2} \right) = \left( a + b + c \right)^2\]
In \[∆ ABC, \frac{b + c}{12} = \frac{c + a}{13} = \frac{a + b}{15}\] Prove that \[\frac{\cos A}{2} = \frac{\cos B}{7} = \frac{\cos C}{11}\]
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 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.
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 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 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
