Advertisements
Advertisements
प्रश्न
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
Advertisements
उत्तर
`sin((A - B)/2) = ((a - b)/c) cos C/2`
Consider `(a - b)/c`
= `(k sin A - k sin B)/(k sin C)` ....(Sine rule)
= `(sin A - sin B)/sin C`
= `(2cos((A + B)/2)sin((A - B)/2))/(sin C)`
= `(2sin C/2 * sin((A - B)/2))/(2sin C/2 cos C/2)` ....(∵ A + B + C = π)
= `sin((A - B)/2)/(cos C/2)`
∴ `(a - b)/2 = sin((A - B)/2)/(cos C/2)`
∴ `sin ((A - B)/2) = ((a - b)/c) cos C/2`
Hence proved.
संबंधित प्रश्न
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`
In any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.
(A) `1/5`
(B) `sqrt(1/5)`
(C) `4/5`
(D) `2/5`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
In , ΔABC prove that
`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`
Find the Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/2)`
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.
`(3/2, (3√3)/2)`.
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
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 Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0
In ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`
In ΔABC, if (a+ b - c)(a + b + c) = 3ab, then ______.
In Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
In a ΔABC, `(sin "C"/2)/(cos(("A" - "B")/2))` = ______
If `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.
In ΔABC, `(sin(B - C))/(sin(B + C))` = ______
In ΔABC if sin2A + sin2B = sin2C and l(AB) = 10, then the maximum value of the area of ΔABC is ______
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
If in Δ ABC, 3a = b + c, then `cot ("B"/2) cot ("C"/2)` = ______.
If in a `triangle"ABC",` a2cos2 A - b2 - c2 = 0, then ______.
If in ΔABC, `sin "B"/2 sin "C"/2 = sin "A"/2` and 2s is the perimeter of the triangle, then s is ______.
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
If in ΔABC, `sin A/2 * sin C/2 = sin B/2` and 2s is the perimeter of the triangle, then s = ______.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
