Advertisements
Advertisements
प्रश्न
With the usual notations, prove that `(sin("A" - "B"))/(sin ("A" + "B")) = ("a"^2 - "b"^2)/"c"^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 = `("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:
sin θ = `-1/2`
Find the principal solution of the following equation:
tan θ = – 1
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation:
tan θ = - 1
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
tan3θ = 3 tanθ.
State whether the following equation have solution or not?
cos 2θ = – 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:
If cos pθ = cos qθ, p ≠ q, then ______.
Select the correct option from the given alternatives:
In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
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 = ______.
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Select the correct option from the given alternatives:
In any ΔABC, if acos B = bcos A, then the triangle is
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
Find the general solutions of the following equation:
sin θ - cos θ = 1
In ΔABC, prove that `("a - b")^2 cos^2 "C"/2 + ("a + b")^2 sin^2 "C"/2 = "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.
Show that `tan^-1 1/2 - tan^-1 1/4 = tan^-1 2/9`.
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 sin x = `-1/2`
`cos^-1 (cos (4pi)/3)` = ______.
`int 1/(sin x * cos x)` dx = ?
If tan-1 x + 2cot-1 x = `(5pi)/6`, then x is
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 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 ______
If sin θ + cos θ = 1, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
The number of principal solutions of tan 2θ = 1 is ______.
Find the general solution of sin θ + sin 3θ + sin 5θ = 0
The general solution to cos100x – sin100x = 1 is ______.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
