Advertisements
Advertisements
प्रश्न
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
Advertisements
उत्तर
Consider that for Δ ABC , ∠B is in a standard position i.e. vertex B is at the origin and the side BC is along positive x-axis. As ∠B is an angle of a triangle ∴ ∠B can be acute or ∠B can be obtuse.
Using the Cartesian co-ordinate system in above figure.
we get B ≡(0,0) A ≡(c cos B , c sin B) and C ≡ (a,0)
Now consider l(CA) = b
∴ b2 = (a- c cos B)2 + (0-c sin B)2 , by distance formula
∴ b2 = a2 - 2 ac cos B + c2 cos2 B +c2 sin2 B
∴ b2 =a2 -2 ac cos B +c2 (sin2B + cos2B)
∴ b2 = a2 +c2 - 2 ac cos B
Hence proved.
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`
In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
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 , ΔABC prove that
`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/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 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))`
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, 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, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
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.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
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, if (a+ b - c)(a + b + c) = 3ab, then ______.
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 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 ______
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)` = ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
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 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 = ______.
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 ______.
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In any ΔABC, prove that:
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.
