If A + B + C = π, prove that
$\sin A+\sin B+\sin C=4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
= sinA + sinB + sinC
Using,
$\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$
$=\sin A+\left\{2 \sin \left(\frac{B+C}{2}\right) \cos \left(\frac{B-C}{2}\right)\right\}$
Since $A+B+C=\pi$
$\rightarrow \mathrm{B}+\mathrm{C}=180-\mathrm{A}$
And,
$\sin \left(\frac{\pi}{2}-\mathrm{A}\right)=\cos \mathrm{A}$
$=\sin A+\left\{2 \sin \left(\frac{\pi-A}{2}\right) \cos \left(\frac{B-C}{2}\right)\right\}$
$=\sin A+\left\{2 \cos \left(\frac{A}{2}\right) \cos \left(\frac{B-C}{2}\right)\right\}$
Using, $\sin 2 A=2 \sin A \cos A$
$=2 \sin \frac{A}{2} \cos \frac{A}{2}+\left\{2 \cos \left(\frac{A}{2}\right) \cos \left(\frac{B-C}{2}\right)\right\}$
$=2 \cos \frac{A}{2}\left\{\sin \frac{A}{2}+\cos \left(\frac{B-C}{2}\right)\right\}$
$\rightarrow \mathrm{B}+\mathrm{C}=180-\mathrm{A}$
And
$\sin \left(\frac{\pi}{2}-A\right)=\cos A$
$=2 \cos \frac{A}{2}\left\{\cos \left(\frac{B+C}{2}\right)+\cos \left(\frac{B-C}{2}\right)\right\}$
$=2 \cos \frac{A}{2}\left\{2 \cos \left(\frac{B}{2}\right) \cos \left(\frac{C}{2}\right)\right\}$
$=4 \cos \frac{A}{2} \cos \left(\frac{B}{2}\right) \cos \left(\frac{C}{2}\right)$
= R.H.S
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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,