Advertisements
Advertisements
प्रश्न
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "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" + "b")/"c") sin "C"/2`
`= (("k sin A" + "k sin B")/("k sin C")) sin "C"/2`
`= (("sin A" + "sin B")/"sin C") sin "C"/2`
`= (2sin (("A" + "B")/2). cos ("A" - "B")/2)/(2"sin" "C"/2. cos "C"/2). sin "C"/2`
`= ("sin" ("A" + "B")/2. cos ("A" - "B")/2)/(cos "C"/2)`
`= (sin (pi/2 - "C"/2).cos ("A" - "B")/2)/(cos "C"/2) ...[because "A" + "B" + "C" = pi]`
`= (cos "C"/2.cos ("A" - "B")/2)/(cos "C"/2)`
`= cos (("A" - "B")/2)`
= LHS
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
Sec θ = `(2)/sqrt(3)`
Find the general solution of the following equation:
tan θ = `(1)/(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:
tan θ = - 1
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
sin θ = tan θ
Find the general solution of the following equation:
tan3θ = 3 tanθ.
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation has a solution or not?
2sinθ = 3
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
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
Select the correct option from the given alternatives:
The principal solutions of equation cot θ = `sqrt3` are ______.
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 if c2 + a2 – b2 = ac, then ∠B = ____.
The principal value of sin–1 `(- sqrt3/2)` is ______.
The principal value branch of sec-1x is ______.
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 principal solutions of the following equation:
cot θ = 0
Find the general solutions of the following equation:
`tan^2 theta = 3`
In ΔABC, prove that `("a - b")^2 cos^2 "C"/2 + ("a + b")^2 sin^2 "C"/2 = "c"^2`
Show that `cos^-1 sqrt3/2 + 2 sin^-1 sqrt3/2 = (5pi)/6`.
If x, y, z are positive, then prove that
`tan^-1 (("x - y")/(1 + "xy")) + tan^-1 (("y - z")/(1 + "yz")) + tan^-1 (("z - x")/(1 + "zx")) = 0`
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 = 0
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
The value of tan 57°- tan 12°- tan 57° tan 12° 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.
Which of the following equations has no solution?
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
The value of sin 18° is ______.
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The general solution of sin 2x = cos 2x is ______
If `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
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 the equation tan2 x = 1 is ______.
Prove that the general solution of cos θ = cos α is θ = 2nπ ± α, n ∈ Z.
If `sin^-1 4/5 + cos^-1 12/13` = sin–1α, then find the value of α.
