Advertisements
Advertisements
प्रश्न
In ΔABC, prove that `sin(("B" − "C")/2) = (("b" − "c")/"a")cos "A"/(2)`.
Advertisements
उत्तर
By the sine rule,
`"a"/"sin A" = "b"/"sin B" = "c"/"sin C"` = k
∴ a = k sin A, b = k sin B, c = k sin C
RHS = `(("b" − "c")/"a")cos "A"/(2)`
RHS = `(("k sin B" − "k sin C")/("k sin A"))cos "A"/(2)`
RHS = `(("sin B" − "sin C")/("sin A"))cos "A"/(2)`
RHS = `(2cos(("B" + "C")/2).sin(("B" − "C")/2))/(2sin "A"/(2) . cos "A"/(2)).cos "A"/(2)`
RHS = `(cos(("B" + "C")/(2)).sin(("B" − "C")/(2)))/(sin "A"/(2)`
RHS = `(cos(π/2 − "A"/2).sin(("B" − "C")/(2)))/(sin "A"/(2))` ...[ ∵ A + B + C = π ]
RHS = `(sin "A"/(2). sin(("B" − "C")/(2)))/(sin "A"/(2)`
RHS = `sin(("B" − "C")/(2))`
LHS = `sin(("B" − "C")/(2))`
RHS = LHS
`sin(("B" − "C")/2) = (("b" − "c")/"a")cos "A"/(2)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
cosθ = `(1)/(2)`
Find the principal solution of the following equation:
Sec θ = `(2)/sqrt(3)`
Find the general solution of the following equation:
cot θ = 0.
Find the general solution of the following equation:
sec θ = `sqrt(2)`.
Find the general solution of the following equation:
cosec θ = - √2.
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
cos 4θ = cos 2θ
Find the general solution of the following equation:
tan3θ = 3 tanθ.
State whether the following equation has a solution or not?
cos2θ = – 1.
In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
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 principal solutions of equation sin θ = `- 1/2` are ______.
Select the correct option from the given alternatives:
The general solution of sec x = `sqrt(2)` is ______.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
The principal value of sin–1 `(- sqrt3/2)` is ______.
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
`cos[tan^-1 1/3 + tan^-1 1/2]` = ______
Find the general solutions of the following equation:
`tan theta = - sqrt3`
Show that `tan^-1 1/2 - tan^-1 1/4 = tan^-1 2/9`.
Prove the following:
`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of sin x − 1 = 0
Find the principal solutions of cos 2𝑥 = 1
Find the principal solutions of sin x = `-1/2`
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
`cos^-1 (cos (4pi)/3)` = ______.
If `|bar"a"|` = 10, `|bar"b"| = 2`, then `sqrt(|bar"a" xx bar"b"|^2 + |bar"a"*bar"b"|^2)` = ?
The general solution of sec θ = `sqrt2` is
If f(x) = sin-1`(sqrt((1 - x)/2))`, then f'(x) = ?
If x + y = `pi/2`, then the maximum value of sin x. sin y is.
The principal solutions of cot x = `sqrt3` are ______.
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The general solution of sin 2x = cos 2x is ______
Which of the following is true in a triangle ABC?
The number of solutions of sin x + sin 3x + sin 5x = 0 in the interval `[pi/2, (3pi)/2]` is ______.
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
If y = `(2sinα)/(1 + cosα + sinα)`, then value of `(1 - cos α + sin α)/(1 + sin α)` is ______.
The general solution of x(1 + y2)1/2 dx + y(1 + x2)1/2 dy = 0 is ______.
The general solution of sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x is ______.
Prove that the general solution of cos θ = cos α is θ = 2nπ ± α, n ∈ Z.
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
