Find the general solution of each of the following equations:
(i) $\sin 2 x=\frac{1}{2}$
(ii) $\cos 3 \mathrm{x}=\frac{1}{\sqrt{2}}$
(iii) $\tan \frac{2 \mathrm{x}}{3}=\sqrt{3}$
To Find: General solution.
(i) Given: $\sin 2 x=\frac{1}{2}$
Formula used: $\sin \theta=\sin \alpha \Rightarrow \theta=n \pi+(-1)^{n} \alpha, n \in I$
By using above formula, we have
$\sin 2 x=\frac{1}{2}=\sin \frac{\pi}{6} \Longrightarrow 2 x=n \pi+(-1)^{n} \cdot \frac{\pi}{6} \Longrightarrow x=\frac{n \pi}{2}+(-1)^{n} \cdot \frac{\pi}{12}, n \in I$
So general solution is $x=\frac{n \pi}{2}+(-1)^{n} \cdot \frac{\pi}{12}$ where $n \in I$
(ii) Given: $\cos 3 x=\frac{1}{\sqrt{2}}$
Formula used: $\cos \theta=\cos \alpha \Rightarrow \theta=2 n \pi \pm \alpha, n \in I$
By using above formula, we have
$\cos 3 x=\frac{1}{\sqrt{2}}=\cos \left(\frac{\pi}{4}\right) \Rightarrow 3 x=2 n \pi \pm \frac{\pi}{4} \Rightarrow x=\frac{2 n \pi}{3} \pm \frac{\pi}{12}, n \in I$
So the general solution is $x=\frac{2 n \pi}{3} \pm \frac{\pi}{12}$ where $n \in$ ।
(iii) Given: $\tan \frac{2 x}{3}=\sqrt{3}$
Formula used: $\tan \theta=\tan \alpha \Rightarrow \theta=\mathrm{n} \pi+\alpha, \mathrm{n} \in$ ।
By using above formula, we have
$\tan \frac{2 \mathrm{x}}{3}=\sqrt{3}=\tan \frac{\pi}{3} \Rightarrow \frac{2 \mathrm{x}}{3}=\mathrm{n} \pi+\frac{\pi}{3} \Rightarrow \mathrm{x}=\frac{3 \mathrm{n} \pi}{2}+\frac{\pi}{2}, \mathrm{n} \in I$
So general solution is $x=(3 n+1) \frac{\pi}{2}$, where $n \in I$
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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,