Understand the problem

Let \(a,b,c\) be the sides of a triangle and \(A,B,C\) be the angles opposite to those sides respectively. If \( \sin (A-B)=\frac{a}{a+b}\sin A\cos B-\frac{b}{a+b} \cos A \sin B\), then prove that the triangle is isosceles.
Source of the problem
I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 6 from 2016
Topic
Trigonometry

Difficulty Level
7 out of 10

Suggested Book
Plane Trigonometry by S.L. Loney

Start with hints

Do you really need a hint? Try it first!

Let \(a,b,c\) be the sides of a triangle and \(A,B,C\) be the angles opposite to those sides respectively. Given \( \sin (A-B)=\frac{a}{a+b}\sin A\cos B-\frac{b}{a+b} \cos A \sin B\) \(\Rightarrow \sin A\cos B- \cos A \sin B=\frac{a}{a+b}\sin A\cos B-\frac{b}{a+b} \cos A \sin B\) \(\Rightarrow \sin A\cos B-\frac{a}{a+b}\sin A\cos B=\cos A \sin B-\frac{b}{a+b} \cos A \sin B\)    

\(\Rightarrow \sin A\cos B-\frac{a}{a+b}\sin A\cos B=\cos A \sin B-\frac{b}{a+b} \cos A \sin B\) \(\Rightarrow \sin A\cos B(1-\frac{a}{a+b})=\cos A \sin B(1-\frac{b}{a+b})\) \(\Rightarrow \frac{b}{a+b}\sin A\cos B=\frac{a}{a+b}\cos A \sin B\)  

\(\Rightarrow \frac{b}{a+b}\sin A\cos B=\frac{a}{a+b}\cos A \sin B\) \(\Rightarrow b\sin A\cos B=a\cos A \sin B\) \(\Rightarrow (\frac{b}{\sin B})(\frac{\cos B}{\cos A})=\frac{a}{\sin A} \)

 

\(\Rightarrow (\frac{b}{\sin B})(\frac{\cos B}{\cos A})=\frac{a}{\sin A} \) \(\Rightarrow (\frac{b}{\sin B})(\frac{\cos B}{\cos A})=\frac{b}{\sin B}\) [since \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)] \(\Rightarrow \frac{\cos B}{\cos A}=1\) \(\cos B=\cos A\) \(A=B\).

\(\Rightarrow \Delta ABC\) is isosceles (Proved).

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Similar Problems

Ratio with the sum of digits

Understand the problemFind the 3-digit number whose ratio with the sum of its digits is minimal.Albania TST 2013 Number Theory, Inequalities. Easy Problem Solving Strategies by Arthur Engel Start with hintsDo you really need a hint? Try it first!Suppose that the...

Does there exist a Magic Rectangle?

Magic Squares are infamous; so famous that even the number of letters on its Wikipedia Page is more than that of Mathematics itself. People hardly talk about Magic Rectangles. Ya, Magic Rectangles! Have you heard of it? No, right? Not me either! So, I set off to...

Coincident Nine-point Circles

Understand the problemLet be a triangle, its circumcenter, its centroid, and its orthocenter. Denote by , and the centers of the circles circumscribed about the triangles , and , respectively. Prove that the triangle is congruent to the triangle and that the...

An inequality with unit coefficients

Understand the problemLet be positive real numbers. Show that there exist such that:Iberoamerican olympiad 2011InequalitiesEasyInequalities: An Approach Through Problems by B.J. VenkatachalaStart with hintsDo you really need a hint? Try it first!Try using...

An integer with perfect square digits

Understand the problemFind all positive integers that have 4 digits, all of them perfect squares, and such that is divisible by 2, 3, 5 and 7. Centroamerican olympiad 2016 Number theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try...

Euler’s theorem and an inequality

Understand the problemLet be the circumcenter and be the centroid of a triangle . If and are the circumcenter and incenter of the triangle, respectively,prove thatBalkan MO 1996 Geometry Easy Let $latex I$ be the incentre. Euler's theorem says that $latex...

A trigonometric relation and its implication

Understand the problemProve that a triangle is right-angled if and only ifVietnam National Mathematical Olympiad 1981TrigonometryMediumChallenge and Thrill of Pre-college MathematicsStart with hintsDo you really need a hint? Try it first!Familiarity with the...

A polynomial equation and roots of unity

Understand the problemFind all polynomials with real coefficients such that divides .Indian National Mathematical Olympiad 2018AlgebraEasyProblem Solving Strategies by Arthur EngelStart with hintsDo you really need a hint? Try it first!Show that, given a zero of...

Bicentric quadrilaterals inside a triangle

Understand the problemLet be a triangle and be cevians concurrent at a point . Suppose each of the quadrilaterals and has both circumcircle and incircle. Prove that is equilateral and coincides with the center of the triangle.Indian team selection test...

Trilinear coordinates and locus

Understand the problemLet be an equilateral triangle and in its interior. The distances from to the triangle's sides are denoted by respectively, where . Find the locus of the points for which can be the sides of a non-degenerate triangle.Romanian Master in...