English

With Usual Notations, in δAbc, Prove that A(B Cos C − C Cos B) = B2 − C2 - Mathematics and Statistics

Advertisements
Advertisements

Question

With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2

Solution

L.H.S. = a(b cos C − c cos B)

= R.H.S.

 

shaalaa.com
  Is there an error in this question or solution?
2012-2013 (October)

APPEARS IN

RELATED QUESTIONS

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 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`


 

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a +   c - b)

 

Find the Cartesian co-ordinates of the point whose polar co-ordinates are:

`(sqrt(2), pi/4)`


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:

`(1/2, (7pi)/3)`


Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

`(0, 1/2)`


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:

`"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2`


In any Δ ABC, prove the following:

`("b" - "c")/"a" = (tan  "B"/2 - tan  "C"/2)/(tan  "B"/2 +tan  "C"/2)`


In Δ ABC, if a, b, c are in A.P., then show that cot `"A"/2, cot  "B"/2, cot  "C"/2` are also in A.P.


In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.


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.


With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C


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.`


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 b2 + c2 − a2 = bc, then ∠A = ______.


If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______


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))`


In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B


In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2 


In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B


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 `sin  ((A - B)/2) = ((a - b)/c) cos  C/2` 


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 `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 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 ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`


In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?


In a ΔABC, cot `(("A - B")/2)* tan (("A + B")/2)` is equal to


In a ΔABC, c2 sin 2B + b2 sin 2C = ?


In a ΔABC if 2 cos C = sin B · cosec A, then ______.


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 Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.


In Δ ABC; with usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.


The polar co-ordinates of P are `(2, pi/6)`. If Q is the image of P about the X-axis then the polar co-ordinates of Q are ______.


In Δ ABC, with the usual notations, if `(tan  "A"/2)(tan  "B"/2) = 3/4` then a + b = ______.


In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______ 


If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______ 


In ΔABC, a = 7cm, b = 3cm and c = 8 cm, then angle A is ______ 


The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.


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 ΔABC, `sin  "B"/2 sin  "C"/2 = sin  "A"/2` and 2s is the perimeter of the triangle, then s 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 `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))`, then a2, b2, c2 are in ______.


In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.


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  ______.


In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.


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)` = ______.


If in ΔABC, `sin  A/2 * sin  C/2 = sin  B/2` and 2s is the perimeter of the triangle, then s = ______.


In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find 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’.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×