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 general solution of the following equation:
sinθ = `1/2`.
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:
4sin2θ = 1.
State whether the following equation has a solution or not?
2sinθ = 3
State whether the following equation have solution or not?
3 tanθ = 5
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
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 ______.
Select the correct option from the given alternatives:
If polar coordinates of a point are `(2, pi/4)`, then its cartesian coordinates are
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
Select the correct option from the given alternatives:
The value of cot (tan-12x + cot-12x) is
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Select the correct option from the given alternatives:
`2 "tan"^-1 (1/3) + "tan"^-1 (1/7) =` _____
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
State whether the following equation has a solution or not?
3 sin θ = 5
Show that `cot^-1 1/3 - tan^-1 1/3 = cot^-1 3/4`.
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`
If | x | < 1, then prove that
`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
Find the principal solutions of cos 2𝑥 = 1
`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)` = ?
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The value of `tan^-1 1/3 + tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/8` is ______.
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The value of θ in (π, 2π) satisfying the equation sin2θ - cos2θ = 1 is ______
The measure of the angle between lines (sin2θ - 1)x2 - 2xy cos2θ + cos2θy2 = 0 is ______
Which of the following is true in a triangle ABC?
If sin θ + cos θ = 1, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
The general solution of cosec x = `-sqrt2` is ______
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
The general solution of the equation tan θ + tan 4θ + tan 7θ = tan θ tan 4θ tan 7θ is ______.
Which of the following equation has no solution?
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 ______.
The general solution to cos100x – sin100x = 1 is ______.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
If tan3θ = cotθ, then θ =
