English

In any Δ ABC, prove the following: a sin A - b sin B = c sin (A - B) - Mathematics and Statistics

Advertisements
Advertisements

Question

In any Δ ABC, prove the following:

a sin A - b sin B = c sin (A - B)

Sum
Advertisements

Solution

By sine rule,

`"a"/("sin A") = "b"/("sin B") = "c"/("sin C") = "k"`

∴ a = k sin A, b = k sin B, c = k sin C

LHS =  a sin A - b sin B

= k sin A. sin A - k sin B. sin B

= k (sin2 A - sin2 B)

= k (sin A + sin B)(sin A - sin B)

`= "k" xx 2 sin  (("A + B")/2). cos (("A - B")/2) xx 2 cos (("A + B")/2). sin (("A - B")/2)`

`= "k" xx 2 sin  (("A + B")/2). cos (("A + B")/2) xx 2 sin (("A - B")/2). cos (("A - B")/2)`

= k × sin (A + B) × sin (A - B)

= k sin (π - C). sin (A - B)  … [∴ A + B + C = π]

= k sin C. sin (A - B)

= c sin (A - B)

= RHS.

shaalaa.com
  Is there an error in this question or solution?
Chapter 3: Trigonometric Functions - Miscellaneous exercise 3 [Page 109]

APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) [English] Standard 12 Maharashtra State Board
Chapter 3 Trigonometric Functions
Miscellaneous exercise 3 | Q 11.1 | Page 109

RELATED QUESTIONS

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 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:

`(3/4, (3pi)/4)`


Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.


In any Δ ABC, prove the following:

a2 sin (B - C) = (b2 - c2) sin A.


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.


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.


In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.


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 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 `(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 ∆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 triangle ABC, If `(sin "A" - sin "C")/(cos "C" - cos "A")` = cot B, then A, B, C are in ________.


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


In a ΔABC, `(sin  "C"/2)/(cos(("A" - "B")/2))` = ______ 


In a ΔABC, 2ab sin`((A + B - C)/2)` = ______


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 `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.


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 ______.


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


In ΔABC, `(sin(B - C))/(sin(B + C))` = ______


In any triangle ABC, the simplified form of `(cos2A)/a^2 - (cos2B)/b^2` 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 ______.


In ΔABC, if `"a" cos^2  "C"/2 + "c" cos^2  "A"/2 = (3"b")/2`, then a, b, c are in ______.


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  ______.


In triangle ABC, a = 4, b = 3 and ∠A = 60°. If ' c' is a root of the equation c2 – 3c – k = 0. Then k = ______. (with usual notations)


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


In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.


If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.


In ΔABC, `(a - b)^2 cos^2  C/2 + (a + b)^2 sin^2  C/2` is equal to ______.


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


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 =


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×