Advertisements
Advertisements
Question
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
Advertisements
Solution
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
RELATED QUESTIONS
Find the principal solution of the following equation:
tan θ = – 1
Find the general solution of the following equation:
sinθ = `1/2`.
Find the general solution of the following equation :
cosθ = `sqrt(3)/(2)`
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:
sin 2θ = `1/2`
Find the general solution of the following equation:
tan3θ = 3 tanθ.
State whether the following equation has a solution or not?
2sinθ = 3
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 `sqrt3`cos x - sin x = 1, then general value of x is ______.
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
The principal value branch of sec-1x is ______.
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the principal solutions of the following equation:
cot θ = 0
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
Show that `cos^-1 sqrt3/2 + 2 sin^-1 sqrt3/2 = (5pi)/6`.
Prove the following:
`cos^-1 "x" = 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.
Find the principal solutions of tan x = `-sqrt(3)`
`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 f(x) = sin-1`(sqrt((1 - x)/2))`, then f'(x) = ?
The number of solutions of `sin^2 theta = 1/2` in [0, π] is ______.
The value of `tan^-1 1/3 + tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/8` 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 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 `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
The general solution of x(1 + y2)1/2 dx + y(1 + x2)1/2 dy = 0 is ______.
Which of the following equation has no solution?
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 ______.
Prove that the general solution of cos θ = cos α is θ = 2nπ ± α, n ∈ Z.
The general solution of cot 4x = –1 is ______.
