In a ∆ABC, if $\frac{\cos A}{a}=\frac{\cos B}{b}$ show that the triangle is isosceles.
Given: $\frac{\cos \mathrm{A}}{\mathrm{a}}=\frac{\cos \mathrm{B}}{\mathrm{b}}$
Need to prove: $\triangle \mathrm{ABC}$ is isosceles.
$\frac{\cos A}{a}=\frac{\cos B}{b}$
$\Rightarrow \frac{\sqrt{1-\sin ^{2} A}}{a}=\frac{\sqrt{1-\sin ^{2} B}}{b}$
$\Rightarrow \frac{1-\sin ^{2} \mathrm{~A}}{\mathrm{a}^{2}}=\frac{1-\sin ^{2} \mathrm{~B}}{\mathrm{~b}^{2}}$ [Squaring both sides]
$\Rightarrow \frac{1}{a^{2}}-\frac{\sin ^{2} A}{a^{2}}=\frac{1}{b^{2}}-\frac{\sin ^{2} B}{b^{2}}$
We know, $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}=2 \mathrm{R}$
Therefore, $\frac{\sin ^{2} \mathrm{~A}}{\mathrm{a}^{2}}=\frac{\sin ^{2} \mathrm{~B}}{\mathrm{~b}^{2}}$
So,
$\Rightarrow \frac{1}{a^{2}}=\frac{1}{b^{2}}$
$\Rightarrow a=b$
That means a and b sides are of same length. Therefore, the triangle is isosceles.
[Proved]
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,