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.
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:
sin θ = `-1/2`
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:
cot θ = 0.
Find the general solution of the following equation:
cosec θ = - √2.
Find the general solution of the following equation:
tan `(2θ)/(3) = sqrt3`.
Find the general solution of the following equation:
cot 4θ = – 1
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
Select the correct option from the given alternatives:
If tan-1(2x) + tan-1(3x) = `pi/4`, then x = _____
Find the general solutions of the following equation:
`tan^2 theta = 3`
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
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`.
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
Prove the following:
`cos^-1 "x" = 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 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`
Find the principal solutions of tan x = `-sqrt(3)`
`int 1/(sin x * cos x)` dx = ?
If `|bar"a"|` = 10, `|bar"b"| = 2`, then `sqrt(|bar"a" xx bar"b"|^2 + |bar"a"*bar"b"|^2)` = ?
`int (sin (log x))^2/x` log x dx = ?
If 2 cos2 θ + 3 cos θ = 2, then permissible value of cos θ is ________.
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
The measure of the angle between lines (sin2θ - 1)x2 - 2xy cos2θ + cos2θy2 = 0 is ______
The general solution of the equation tan θ tan 2θ = 1 is given by ______
The general solution of 4sin2 x = 1 is ______.
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
The number of solutions of sin x + sin 3x + sin 5x = 0 in the interval `[pi/2, (3pi)/2]` 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 ______.
The general solution of cot 4x = –1 is ______.
