An isosceles triangle of vertical angle $2 \theta$ is inscribed in a circle of radius $a$. Show that the area of the triangle is maximum when $\theta=\frac{\pi}{6}$.
Let $\mathrm{ABC}$ be an isosceles triangle inscribed in the circle with radius a such that $\mathrm{AB}=\mathrm{AC}$.
$\mathrm{AD}=\mathrm{AO}+\mathrm{OD}=a+a \cos 2 \theta=a(1+\cos 2 \theta)$ and
$\mathrm{BC}=2 \mathrm{BD}=2 a \sin 2 \theta$
As, area of the triangle $\mathrm{AC}, A=\frac{1}{2} \mathrm{BC} \times \mathrm{AD}$
$\Rightarrow A(\theta)=\frac{1}{2} \times 2 a \sin 2 \theta \times a(1+\cos 2 \theta)$
$=a^{2} \sin 2 \theta(1+\cos 2 \theta)$
$=a^{2} \sin 2 \theta+a^{2} \sin 2 \theta \cos 2 \theta$
$\Rightarrow A(\theta)=a^{2} \sin 2 \theta+\frac{a^{2} \sin 4 \theta}{2}$
$\Rightarrow A^{\prime}(\theta)=2 a^{2} \cos 2 \theta+\frac{4 a^{2} \cos 4 \theta}{2}$
$\Rightarrow A^{\prime}(\theta)=2 a^{2} \cos 2 \theta+2 a^{2} \cos 4 \theta$
$\Rightarrow A^{\prime}(\theta)=2 a^{2}(\cos 2 \theta+\cos 4 \theta)$
For maxima or minima, $A^{\prime}(\theta)=0$
$\Rightarrow 2 a^{2}(\cos 2 \theta+\cos 4 \theta)=0$
$\Rightarrow \cos 2 \theta+\cos 4 \theta=0$
$\Rightarrow \cos 2 \theta=-\cos 4 \theta$
$\Rightarrow \cos 2 \theta=\cos (\pi-4 \theta)$
$\Rightarrow 2 \theta=\pi-4 \theta$
$\Rightarrow 6 \theta=\pi$
$\Rightarrow \theta=\frac{\pi}{6}$
Also, $A^{\prime \prime}(\theta)=2 a^{2}(-\sin 2 \theta-\sin 4 \theta)=-2 a^{2}(\sin 2 \theta+\sin 4 \theta)<0$ at $\theta=\frac{\pi}{6}$
So, the area of the triangle is maximum at $\theta=\frac{\pi}{6}$.
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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,