Write the value of $\sin \frac{\pi}{15} \sin \frac{4 \pi}{15} \sin \frac{3 \pi}{10}$
$\frac{\pi}{15}=12^{\circ}, \frac{4 \pi}{15}=48^{\circ}, \frac{3 \pi}{10}=54^{\circ}$
$\sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}$
$=\frac{1}{2}\left[2 \sin 12^{\circ} \sin 48^{\circ}\right] \sin 54^{\circ}$
$=\frac{1}{2}\left[\cos \left(12^{\circ}-48^{\circ}\right)-\cos \left(12^{\circ}+48^{\circ}\right)\right] \sin 54^{\circ}$
$=\frac{1}{2}\left[\cos \left(-36^{\circ}\right)-\cos 60^{\circ}\right] \sin 54^{\circ}$
$=\frac{1}{2} \sin 54^{\circ}\left[\cos 36^{\circ}-\frac{1}{2}\right]$
$=\frac{1}{2}\left[\sin \left(90^{\circ}-36^{\circ}\right) \cos 36^{\circ}\right]-\frac{1}{4} \sin \left(90^{\circ}-36^{\circ}\right)$
$=\frac{1}{2} \cos ^{2} 36^{\circ}-\frac{1}{4} \cos 36^{\circ}$
$=\frac{1}{2}\left(\frac{\sqrt{5}+1}{4}\right)^{2}-\left(\frac{\sqrt{5}+1}{16}\right) \quad\left[\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}\right]$
$=\frac{1}{2}\left(\frac{5+1+2 \sqrt{5}}{16}\right)-\left(\frac{\sqrt{5}+1}{16}\right)$
$=\frac{6+2 \sqrt{5}}{32}-\frac{\sqrt{5}+1}{16}$
$=\frac{6+2 \sqrt{5}-2 \sqrt{5}-2}{32}$
$=\frac{4}{32}$
$=\frac{1}{8}$
- 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,
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,