Advertisements
Advertisements
प्रश्न
In , ΔABC with usual notations prove that
(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`
Advertisements
उत्तर
Taking LHS
= `("a"-"b")^2 "cos"^2 "C"/2 + ("a"+"b")^2 "sin"^2 "C"/2`
`=("a"^2+"b"^2 -2"ab") "cos"^2"C"/2 +("a"^2 +"b"^2 +2"ab")."sin""C"/2`
`= ("a"^2+"b"^2)"cos""C"/2 - 2 "ab" "cos"^2"C"/2 + ("a"^2+"b"^2) . "sin"^2"C" /2 + 2 "ab" "sin"^2 "C"/2`
`=("a"^2 + "b"^2)("cos"^2"C"/2 +"sin"^2"C"/2) - 2"ab"("cos"^2"C"/2 - "sin"^2"C"/2)`
=`("a"^2 +"b"^2) -2"ab" "cos""C"` {By cosine Rule}
= c2
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 any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(3/4, (3pi)/4)`
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)`
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
In any Δ ABC, prove the following:
a sin A - b sin B = c sin (A - B)
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 sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B
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 `(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`
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, if (a+ b - c)(a + b + c) = 3ab, then ______.
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 `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.
In Δ ABC; with usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.
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))` = ______
If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
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 = ______.
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 `("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)`.
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, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
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, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
