Advertisements
Advertisements
प्रश्न
In ΔABC, if a cos A = b cos B then prove that the triangle is either a right angled or an isosceles traingle.
Advertisements
उत्तर
Using the sine rule,
`a/"sinA" = b/"sinB"` = k
a = k sin A and b = k sin B
∴ a cos A = b cos B gives
k sinA cosA = k sinB cosB
∴ 2sinA cosA = 2sinB cosB
∴ sin 2A = sin 2B
∴ sin2A – sin2B = 0
∴ 2cos(A + B).sin(A – B) = 0
∴ 2cos(π – C).sin(A – B) = 0 ...[∵ A + B + C = π]
∴ - 2cosC. sin(A – B) = 0
∴ cosC = 0 OR sin(A – B) = 0
∴ C = 90° OR A – B = 0
∴ C = 90° OR A = B
∴ the triangle is either rightangled or an isosceles triangle.
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
cosθ = `(1)/(2)`
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation:
sinθ = `1/2`.
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:
tan `(2θ)/(3) = sqrt3`
Find the general solution of the following equation:
4sin2θ = 1.
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 have solution or not?
3 tanθ = 5
In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
If `sqrt3`cos x - sin x = 1, then general value of x is ______.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the principal solutions of the following equation:
sin 2θ = `-1/2`
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the general solutions of the following equation:
sin θ - cos θ = 1
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`.
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))`
Find the principal solutions of cos 2𝑥 = 1
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
`cos^-1 (cos (4pi)/3)` = ______.
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If 2 cos2 θ + 3 cos θ = 2, then permissible value of cos θ is ________.
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
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 ______.
The general solution of the equation tan θ tan 2θ = 1 is given by ______
Which of the following is true in a triangle ABC?
The general solution of cot θ + tan θ = 2 is ______.
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
Find the principal solutions of cot θ = 0
The general solution of the equation tan θ + tan 4θ + tan 7θ = tan θ tan 4θ tan 7θ is ______.
General solution of the equation sin 2x – sin 4x + sin 6x = 0 is ______.
With usual notations, in any ΔABC, if a cos B = b cos A, then the triangle 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 ______.
If `2sin^-1 3/7` = cos–1β, then find the value of β.
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
