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 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 ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - 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`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(3/4, (3pi)/4)`
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.
`(3/2, (3√3)/2)`.
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In any Δ ABC, prove the following:
a sin A - b sin B = c sin (A - B)
In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
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.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B
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 `sin ((A - B)/2) = ((a - b)/c) cos C/2`
In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle 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, `(sin(B - C))/(sin(B + C))` = ______
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 ______
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 ______.
If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
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 a ΔABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))`, then a2, b2, c2 are in ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
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 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 ______.
In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
The perimeter of ΔABC is 20, ∠A = 60°, area of ΔABC = `10sqrt(3)`, then find the values of a, b, c.
In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
