Advertisements
Advertisements
प्रश्न
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
Advertisements
उत्तर १
L.H.S. = `2{a sin^2 "C"/2 + "c"sin^2 "A"/2}`
= `2 {a ((1 – cos "C")/2) + c(1 – cos A/2)}`
= `2{(a - a cos "C" + c - c cos"A")}`
= `a + c - (c cos "A" + a cos "C")` `...[∵ "By Projection Rule b" = c cos"A" + a cos "C"]`
= a + c − b
= a - b + c
L. H. S. = R.H.S.
उत्तर २
L.H.S. = `2{a sin^2 "C"/2 + "c"sin^2 "A"/2}`
= `a(2sin^2 "C"/2)+c(2sin^2 "A"/2)`
= a (1 - cos C) + c (1 - cos A)
= `a[1 - (a^2+b^2-c^2)/(2ab)]+c[(b^2+c^2-a^2)/(2bc)]` ...[By cosine rule]
= `a[(2ab-a^2-b^2+c^2)/(2ab)]+c[(2bc-b^2-c^2+a^2)/(2bc)]`
= `(2ab-a^2-b^2+c^2)/(2b)+(2bc-b^2-c^2+a^2)/(2b)`
= `(2ab-a^2-b^2+c^2+2bc-b^2-c^2+a^2)/(2b)`
= `(2ab-2b^2+2bc)/(2b)`
= a - b + c
= RHS
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
cosθ = `(1)/(2)`
Find the principal solution of the following equation:
sin θ = `-1/2`
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation :
cosθ = `sqrt(3)/(2)`
Find the general solution of the following equation:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
tan θ = - 1
Find the general solution of the following equation:
tan `(2θ)/(3) = sqrt3`
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation have solution or not?
cos 2θ = – 1
State whether the following equation have solution or not?
3 tanθ = 5
In ΔABC, prove that `sin(("B" − "C")/2) = (("b" − "c")/"a")cos "A"/(2)`.
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
Select the correct option from the given alternatives:
The general solution of sec x = `sqrt(2)` is ______.
Select the correct option from the given alternatives:
If cos pθ = cos qθ, p ≠ q, then ______.
If `sqrt3`cos x - sin x = 1, then general value of x is ______.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
`"cos"^-1 ("cos" (7pi)/6)` = _________.
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the general solutions of the following equation:
`tan^2 theta = 3`
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
Show that `tan^-1 1/2 = 1/3 tan^-1 11/2`
Show that `cos^-1 sqrt3/2 + 2 sin^-1 sqrt3/2 = (5pi)/6`.
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of cosec x = 2
Find the principal solutions of sin x = `-1/2`
If x + y = `pi/2`, then the maximum value of sin x. sin y is.
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
The number of solutions of `sin^2 theta = 1/2` in [0, π] is ______.
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
The value of sin 18° is ______.
The general solution of the equation tan θ tan 2θ = 1 is given by ______
Find the principal solutions of cot θ = 0
The general solution of the equation tan2 x = 1 is ______.
With usual notations, in any ΔABC, if a cos B = b cos A, then the triangle is ______.
Find the general solution of sin θ + sin 3θ + sin 5θ = 0
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
If tan3θ = cotθ, then θ =
