Advertisements
Advertisements
प्रश्न
In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
Advertisements
उत्तर
`sin^2theta =(1-cos2theta)/2`
`L.H.S=2{asin^2(C/2)+csin^2(A/2)}`
`=2{(a(1-cosC))/2+(c(1-cosA))/2}`
=a-acosC+c-ccosA
=(a+c)-(acosC+ccosA)
=a+c-b
R.H.S
2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
APPEARS IN
संबंधित प्रश्न
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
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 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)`
In any Δ ABC, prove the following:
`("b" - "c")/"a" = (tan "B"/2 - tan "C"/2)/(tan "B"/2 +tan "C"/2)`
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
State whether the following equation has a solution or not?
cos 2θ = `1/3`
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.
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, cot `(("A - B")/2)* tan (("A + B")/2)` is equal to
In a ΔABC if 2 cos C = sin B · cosec A, then ______.
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
In a ΔABC, 2ab sin`((A + B - C)/2)` = ______
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 Δ ABC; with usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.
In ΔABC, `(sin(B - C))/(sin(B + C))` = ______
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
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 ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
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 a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
If in ΔABC, `sin A/2 * sin C/2 = sin B/2` and 2s is the perimeter of the triangle, then s = ______.
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 =
