Advertisements
Advertisements
Question
In , ΔABC with usual notations prove that
(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`
Advertisements
Solution
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
RELATED QUESTIONS
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.
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
(A) `1/sqrt5`
(B) `1/sqrt10`
(C) `1/sqrt15`
(D) `1/(2sqrt5)`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
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:
`(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:
`(1/2, (7pi)/3)`
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
In any Δ ABC, prove the following:
`("b" - "c")/"a" = (tan "B"/2 - tan "C"/2)/(tan "B"/2 +tan "C"/2)`
In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In ΔABC, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a ΔABC if 2 cos C = sin B · cosec A, then ______.
In ΔABC, `(sin(B - C))/(sin(B + C))` = ______
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
In any triangle ABC, the simplified form of `(cos2A)/a^2 - (cos2B)/b^2` 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 ______.
If a = 13, b = 14, c = 15, then `cos("A"/2)` = ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
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 ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, `(a - b)^2 cos^2 C/2 + (a + b)^2 sin^2 C/2` is equal to ______.
