In any ΔABC, prove that
$\frac{(a+b)}{c} \sin \frac{C}{2}=\cos \frac{(A-B)}{2}$
Need to prove: $\frac{(\mathrm{a}+\mathrm{b})}{\mathrm{c}} \sin \frac{\mathrm{C}}{2}=\cos \frac{(\mathrm{A}-\mathrm{B})}{2}$
We know that, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$ where $R$ is the circumradius.
Therefore,
$a=2 R \sin A \cdots$ (a)
Similarly, $b=2 R \sin B$ and $c=2 R \sin C$
Now, $\frac{a+b}{c}=\frac{2 R(\sin A+\sin B)}{2 R \sin C}=\frac{\sin A+\sin B}{\sin C}$
Therefore, $\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=\frac{\sin \mathrm{C}}{\sin \mathrm{A}+\sin \mathrm{B}}=\frac{2 \sin \frac{\mathrm{C}}{2} \cos \frac{\mathrm{C}}{2}}{2 \sin \frac{\mathrm{A}+\mathrm{B}}{2} \cos \frac{\mathrm{A}-\mathrm{B}}{2}}$
$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=\frac{\sin \frac{\mathrm{C}}{2} \cos \frac{\mathrm{C}}{2}}{\sin \left(\frac{\pi}{2}-\frac{\mathrm{C}}{2}\right) \cos \frac{\mathrm{A}-\mathrm{B}}{2}}$
$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=\frac{\sin \frac{\mathrm{C}}{2} \cos \frac{\mathrm{C}}{2}}{\cos \frac{\mathrm{C}}{2} \cos \frac{\mathrm{A}-\mathrm{B}}{2}}$
$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}}=\frac{\sin \frac{\mathrm{C}}{2}}{\cos \frac{\mathrm{A}-\mathrm{B}}{2}}$
$\Rightarrow \frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}} \sin \frac{\mathrm{C}}{2}=\cos \frac{\mathrm{A}-\mathrm{B}}{2}[$ 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,