Advertisements
Advertisements
प्रश्न
In any ΔABC, prove the following:
`("c" - "b cos A")/("b" - "c cos A") = ("cos B")/("cos C")`
Advertisements
उत्तर
L.H.S. = `("c" - "b cos A")/("b" - "c cos A")`
`= ("c" - "b" (("b"^2 + "c"^2 - "a"^2)/(2"bc")))/("b" - "c" (("b"^2 + "c"^2 - "a"^2)/(2"bc")))`
`= ("c" - (("b"^2 + "c"^2 - "a"^2)/"2c"))/("b" - (("b"^2 + "c"^2 - "a"^2)/"2b"))`
`= ((2"c"^2 - "b"^2 - "c"^2 + "a"^2)/"2c")/((2"b"^2 - "b"^2 - "c"^2 + "a"^2)/"2b")`
= `((("c"^2 + "a"^2 - "b"^2)/"2c"))/((("a"^2 + "b"^2 - "c"^2)/"2b")) `
`= ((("c"^2 + "a"^2 - "b"^2)/"2ca"))/((("a"^2 + "b"^2 - "c"^2)/"2ab")`
= `"cos B"/"cos C"`
= R.H.S.
APPEARS IN
संबंधित 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.
In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(sqrt(2), pi/4)`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(1/2, (7pi)/3)`
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.
In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
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)`.
State whether the following equation has a solution or not?
cos 2θ = `1/3`
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
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 ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`
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 ____________.
In a triangle ABC, If `(sin "A" - sin "C")/(cos "C" - cos "A")` = cot B, then A, B, C are in ________.
In a ΔABC, `(sin "C"/2)/(cos(("A" - "B")/2))` = ______
In a ΔABC, 2ab sin`((A + B - C)/2)` = ______
In Δ ABC, with the usual notations, if `(tan "A"/2)(tan "B"/2) = 3/4` then a + b = ______.
If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is ______.
If in a `triangle"ABC",` a2cos2 A - b2 - c2 = 0, then ______.
In a triangle ABC, b = `sqrt3`, c = 1 and ∠A = 30°, then the largest angle of the triangle is ______
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle 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 any ΔABC, prove that:
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.
If the angles A, B, C of a ΔABC are in A.P. and ∠A = 30°, c = 5, then find the values of ‘a’ and ‘b’.
In a triangle ABC, with usual notations, if \[\frac{2\cos\mathrm{A}}{\mathrm{a}}+\frac{\cos\mathrm{B}}{\mathrm{b}}+\frac{2\cos\mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ac}}\]then ∠A =
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
