Write $\left(i^{25}\right)^{3}$ in polar form.
$\left(i^{25}\right)^{3}=i^{75}$
$=i^{4 \times 18+3}$
$=\left(i^{4}\right)^{18} \cdot i^{3}$
$=i^{3} \quad\left[\because i^{4}=1\right]$
$=-i \quad\left[\because i^{3}=-i\right]$
Let $z=0-i$
Then, $|z|=\sqrt{0^{2}+(-1)^{2}}=1$
Let $\theta$ be the argument of $z$ and $\alpha$ be the acute angle given by $\tan \alpha=\frac{|\operatorname{Im}(z)|}{|\operatorname{Re}(z)|}$.
Then,
$\tan \alpha=\frac{1}{0}=\infty$
$\Rightarrow \alpha=\frac{\pi}{2}$
Clearly, $z$ lies in fourth quadrant. So, $\arg (z)=-\alpha=-\frac{\pi}{2}$.
$\therefore$ the polar form of $z$ is $|z|(\cos \theta+i \sin \theta)=\cos \left(-\frac{\pi}{2}\right)+i \sin \left(-\frac{\pi}{2}\right)$.
Thus, the polar form of $\left(i^{25}\right)^{3}$ is $\cos \left(\frac{\pi}{2}\right)-i \sin \left(\frac{\pi}{2}\right)$.
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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,