Advertisements
Advertisements
प्रश्न
In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0
Advertisements
उत्तर
In ∆ABC by sine rule, we have
`(sin"A")/"a" = (sin"B")/"b" = (sin"C")/"c"` = k
∴ sin A = ka, sin B = kb, sin C = kc
Consider a2sin(B − C) = a2(sin B cos C − cos B sin C)
= a2(kb cos C − kc cos B)
= ka(ab cos C − ac cos B)
= `"ak"["ab"(("a"^2 + "b"^2 - "c"^2)/(2"ab")) - "ac"(("a"^2 + "c"^2 - "b"^2)/(2"ac"))]` .......[By consine rule]
= `"ak"[("a"^2 + "b"^2 + "c"^2)/2 - (("a"^2 + "c"^2 - "b"^2)/2)]`
= `"k"/2 ("a")("a"^2 + "b"^2 - "c"^2 - "a"^2 - "c"^2 + "b"^2)`
= `"k"/2 "a"(2"b"^2 - 2"c"^2)`
= ka(b2 − c2)
Similarly, we can prove that
b2sin(C − A) = kb(c2 − a2) and c2sin(A − B)
= kc(a2 − b2)
∴ `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")`
= `("ka"("b"^2 - "c"^2))/(sin"A") + ("kb"("c"^2 - "a"^2))/(sin"B") + ("kc"("a"^2 - "b"^2))/(sin"C")`
= `("ka"("b"^2 - "c"^2))/"ka" + ("kb"("c"^2 - "a"^2))/"kb" + ("kc"("a"^2 - "b"^2))/"kc"`
= (b2 − c2 + c2 − a2 + a2 − b2)
= 0
∴ `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0
संबंधित प्रश्न
If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
(A) `1/sqrt5`
(B) `1/sqrt10`
(C) `1/sqrt15`
(D) `1/(2sqrt5)`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
The principal solutions of cot x = -`sqrt3` are .................
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(1/2, (7pi)/3)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(sqrt(2), sqrt(2))`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(0, 1/2)`
Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.
In Δ ABC, if a, b, c are in A.P., then show that cot `"A"/2, cot "B"/2, cot "C"/2` are also in A.P.
In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
State whether the following equation has a solution or not?
cos 2θ = `1/3`
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.
In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In a triangle ABC with usual notations, if `(cos "A")/"a" = (cos "B")/"b" = (cos "C")/"c"`, then area of triangle ABC with a = `sqrt6` is ____________.
In Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
If one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is ______
If `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.
If P(6, 10, 10), Q(1, 0, -5), R(6, -10, λ) are vertices of a triangle right angled at Q, then value of λ is ______.
The polar co-ordinates of P are `(2, pi/6)`. If Q is the image of P about the X-axis then the polar co-ordinates of Q are ______.
In ΔABC if sin2A + sin2B = sin2C and l(AB) = 10, then the maximum value of the area of ΔABC is ______
If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______
In ΔABC, a = 7cm, b = 3cm and c = 8 cm, then angle A is ______
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
In ΔABC, if `"a" cos^2 "C"/2 + "c" cos^2 "A"/2 = (3"b")/2`, then a, b, c are in ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC is ______.
In triangle ABC, a = 4, b = 3 and ∠A = 60°. If ' c' is a root of the equation c2 – 3c – k = 0. Then k = ______. (with usual notations)
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
The number of solutions of the equation sin 2x – 2 cosx + 4 sinx = 4 in the interval [0, 5π] is ______.
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
If in ΔABC, `sin A/2 * sin C/2 = sin B/2` and 2s is the perimeter of the triangle, then s = ______.
If the angles A, B, C of a ΔABC are in A.P. and ∠A = 30°, c = 5, then find the values of ‘a’ and ‘b’.
