Advertisements
Advertisements
प्रश्न
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
Advertisements
उत्तर
Consider L.H.S. = `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c"`
= `1/"a" (cos "A") + 1/"b" (cos "B") + 1/"c" (cos "C")`
= `1/"a" (("b"^2 + "c"^2 - "a"^2)/(2"bc")) + 1/"b" (("a"^2 + "c"^2 - "b"^2)/(2"ac")) + 1/"c" (("a"^2 + "b"^2 - "c"^2)/(2"ab"))` .......[By consine rule]
= `("b"^2 + "c"^2 - "a"^2)/(2"abc") + ("a"^2 + "c"^2 - "b"^2)/(2"abc") + ("a"^2 + "b"^2 - "c"^2)/(2"abc")`
= `("b"^2 + "c"^2 - "a"^2 + "a"^2 + "c"^2 - "b"^2 + "a"^2 + "b"^2 - "c"^2)/(2"abc")`
= `("a"^2 + "b"^2 + "c"^2)/(2"abc")`
= R.H.S.
∴ `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
APPEARS IN
संबंधित प्रश्न
In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`
In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
The principal solutions of cot x = -`sqrt3` are .................
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
In , ΔABC with usual notations prove that
(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(sqrt(2), pi/4)`
Find the Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/2)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(0, 1/2)`
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:
a sin A - b sin B = c sin (A - B)
In any Δ ABC, prove the following:
ac cos B - bc cos A = a2 - b2
In any Δ ABC, prove the following:
`"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2`
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 ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
In Δ ABC, if a cos2 `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B
In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`
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 a ΔABC, c2 sin 2B + b2 sin 2C = ?
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
If in a right-angled triangle ABC, the hypotenuse AB = p, then `overline"AB".overline" AC" + overline"BC".overline" BA" + overline" CA".overline"CB"` is equal to ______
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 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 ______.
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 ______
In ΔABC, if `"a" cos^2 "C"/2 + "c" cos^2 "A"/2 = (3"b")/2`, then a, b, c are in ______.
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
If a = 13, b = 14, c = 15, then `cos("A"/2)` = ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In a triangle ABC, with usual notations, if \[\frac{2\cos\mathrm{A}}{\mathrm{a}}+\frac{\cos\mathrm{B}}{\mathrm{b}}+\frac{2\cos\mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ac}}\]then ∠A =
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
