Advertisements
Advertisements
प्रश्न
With the usual notations, prove that `(sin("A" - "B"))/(sin ("A" + "B")) = ("a"^2 - "b"^2)/"c"^2`
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 = `("a"^2 - "b"^2)/"c"^2 = ("k"^2 "sin"^2 "A" - "k"^2 "sin"^2 "B")/("k"^2 "sin"^2 "C")`
`= ("sin"^2"A" - "sin"^2"B")/("sin"^2"C")`
`= ((sin "A" + sin "B")(sin "A" - sin "B"))/[sin{pi - ("A" + "B")}]^2` .....[∵ A + B + C = π]
`= (2 sin (("A" + "B")/2). cos (("A" - "B")/2) xx 2 cos (("A" + "B")/2). sin (("A" - "B")/2))/("sin"^2 ("A + B"))`
`= (2 sin (("A" + "B")/2). cos (("A" + "B")/2) xx 2 sin (("A" - "B")/2). cos (("A" - "B")/2))/("sin"^2 ("A + B"))`
`= ("sin" ("A" + "B"). sin("A" - "B"))/("sin"^2 ("A" + "B"))`
`= (sin ("A - B"))/(sin ("A + B"))` = LHS
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation :
cot θ = `sqrt(3)`
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:
4sin2θ = 1.
Find the general solution of the following equation:
tan3θ = 3 tanθ.
State whether the following equation has a solution or not?
cos2θ = – 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
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:
In ΔABC, ac cos B - bc cos A = _______
The principal value of sin–1 `(- sqrt3/2)` is ______.
Select the correct option from the given alternatives:
The principal value branch of sec-1x is
Find the general solutions of the following equation:
`tan theta = - sqrt3`
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
Find the general solutions of the following equation:
sin θ - cos θ = 1
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
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
Prove the following:
`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0
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 `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1
Find the principal solutions of cos 2𝑥 = 1
If `|bar"a"|` = 10, `|bar"b"| = 2`, then `sqrt(|bar"a" xx bar"b"|^2 + |bar"a"*bar"b"|^2)` = ?
If tan-1 x + 2cot-1 x = `(5pi)/6`, then x 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 value of θ in (π, 2π) satisfying the equation sin2θ - cos2θ = 1 is ______
The general solution of sin 2x = cos 2x is ______
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
The general solution of cosec x = `-sqrt2` is ______
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 number of principal solutions of tan 2θ = 1 is ______.
Principal solutions at the equation sin 2x + cos 2x = 0, where π < x < 2 π are ______.
The general solution of the equation tan2 x = 1 is ______.
