Electrical engineers are perhaps the most common users of complex numbers to model physical reality. Start with Ohm's Law, $V=IR$. That works for resistors, which dissipate energy as heat. But there are also capacitors, which store energy in electric fields, and inductors (coils), which store energy in magnetic fields. We may generalize resistance $R$ as impedance $Z$. $Z=1/i\omega C$ for a capacitor, and $Z=i\omega L$ for and inductor ($\omega$ is $2\pi f$, where f is the frequency of a sinusoidal signal component). This turns out to be an extremely useful tool for linear circuit analysis.
sinusoidal modeling algebra 2 with trigonometry homework
It is more elegant (i.e. the math ends up being much simpler) to describe sinusoidal oscillations (these may be physical oscillations of a pendulum or mass on a spring, or temporal or spatial oscillations of the direction or amplitude of a vector field such as the electromagnetic field) in terms of complex numbers rather than sines and cosines. This is because in the former you case you need only keep track of an amplitude and phase and addition and multiplication are simple addition and multiplication of complex numbers. If instead you want to use sines and cosines (i.e. avoid complex numbers) you will find yourself having to memorize and apply numerous trig identities and your algebraic expressions will quickly blow up in complexity. It turns out that differential equations are similarly simplified when moving from sinusoidal to complex mathematical approaches.
Can all physics (including QM) be done without complex numbers? yes. Would the mathematics (addition and multiplication of signals and differential equations) get painful fast because trigonometry is ugly? yes.
This introductory, algebra-based, two-semester college physics book is grounded with real-world examples, illustrations, and explanations to help students grasp key, fundamental physics concepts. This online, fully editable and customizable title includes learning objectives, concept questions, links to labs and simulations, and ample practice opportunities to solve traditional physics application problems.
Vectors - magnitude, direction, component form, trigonometric form, unit vector, algebra of vectors, applications,TMM 002 PRECALCULUS (Revised March 21, 2017)AdditionalOptional Learning Outcomes:2. Geometry: The successful Precalculus student can:2e. Interpret the result of vector computations geometrically and within the confines of a particular applied context (e.g., forces).Sample Tasks:The student can define vectors, their arithmetic, their representation, and interpretations.The student can decompose vectors into normal and parallel components.The student can interpret the result of a vector computation as a change in location in the plane or as the net force acting on an object.
Applied Calculus instructs students in the differential and integral calculus of elementary functions with an emphasis on applications to business, social and life science. Different from a traditional calculus course for engineering, science and math majors, this course does not use trigonometry, nor does it focus on mathematical proofs as an instructional method. 2ff7e9595c
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