Advertisements
Advertisements
प्रश्न
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
उत्तर
Given that a2 ,b2 ,c2 are in arithmetic progression.
We need to prove that cotA, cotB and cotC are in
arithmetic progression.
a2 ,b2 ,c2 are in A.P.
`-2a^2, -2b^2, -2c^2 " are in A.P"`
`(a^2+b^2+c^2)-2a^2,(a^2+b^2+c^2)-2b^2, (a^2+b^2+c^2)-2c^2 " are in A.P"`
`(b^2+c^2-a^2), (c^2+a^2-b^2),(a^2+b^2-c^2) " are in A.P "`
`(b^2+c^2-a^2)/(2abc), (c^2+a^2-b^2)/(2abc),(a^2+b^2-c^2) /(2abc)" are in A.P "`
`1/a(b^2+c^2-a^2)/(2bc), 1/b(c^2+a^2-b^2)/(2ac),1/c(a^2+b^2-c^2) /(2ab)" are in A.P "`
`1/acosA,1/bcosB,1/c cos C " are in A.P"`
`k/acosA,k/bcosB,k/c cos C " are in A.P"`
`cosA/sinA,cosB/sinB,cosC/sinC " are in A.P"`
`cotA,cotB,cotC " are in A.P"`
APPEARS IN
संबंधित प्रश्न
In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
The principal solutions of cot x = -`sqrt3` are .................
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
In , ΔABC with usual notations prove that
(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`
Find the Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/2)`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(3/4, (3pi)/4)`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(1/2, (7pi)/3)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(1, - sqrt(3))`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(3/2, (3√3)/2)`.
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In any Δ ABC, prove the following:
ac cos B - bc cos A = a2 - b2
In any Δ ABC, prove the following:
`"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2`
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
In Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
In Δ ABC, if a cos2 `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.
Show that `2 sin^-1 (3/5) = tan^-1(24/7)`
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`
In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled
In ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0
In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In a ΔABC, c2 sin 2B + b2 sin 2C = ?
In a ΔABC if 2 cos C = sin B · cosec A, then ______.
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
In a triangle ABC with usual notations, if `(cos "A")/"a" = (cos "B")/"b" = (cos "C")/"c"`, then area of triangle ABC with a = `sqrt6` is ____________.
If in a right-angled triangle ABC, the hypotenuse AB = p, then `overline"AB".overline" AC" + overline"BC".overline" BA" + overline" CA".overline"CB"` is equal to ______
If one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is ______
If P(6, 10, 10), Q(1, 0, -5), R(6, -10, λ) are vertices of a triangle right angled at Q, then value of λ is ______.
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
If in ΔABC, `sin "B"/2 sin "C"/2 = sin "A"/2` and 2s is the perimeter of the triangle, then s is ______.
In a triangle ABC, b = `sqrt3`, c = 1 and ∠A = 30°, then the largest angle of the triangle is ______
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC is ______.
The number of solutions of the equation sin 2x – 2 cosx + 4 sinx = 4 in the interval [0, 5π] is ______.
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, with usual notations, if a, b, c are in A.P. Then `a cos^2 (C/2) + c cos^2(A/2)` = ______.
In ΔABC, `(a - b)^2 cos^2 C/2 + (a + b)^2 sin^2 C/2` is equal to ______.
In any ΔABC, prove that:
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.
