Represent the following four complex numbers in rectangular form a + ib, polar form \(\left| z \right|e^{i\varphi }\), and on an Argand diagram:\(e^{i\pi }\); i ; \(\sin \left( \pi /2\right)\); \(\cos \left( \pi /4\right)+i\sin \left( -\pi /4 \right)\)

Solve the following complex equation for the unknown quantities \(\left| q_{0} \right|\) and \(\varphi\) in terms of the known quantities a, b, c : \(\left| q_{0} \right|e^{i\varphi }=\frac{a}{b+ic}\)

Euler’s formula \(e^{i\theta }=\cos \theta +i\sin \theta\) is very important and you must be able to write it down without even thinking. Starting from Euler’s formula, find expressions for (i) cosθ and (ii) sinθ in terms of exponential functions. Take as the definitions of the hyperbolic trigonometric functions (iii) \(\cosh \theta \equiv \cos \left( i\theta \right)\) and
(iv) \(\sinh \theta \equiv -i\sin \left( i\theta \right)\). Using the expressions you found in (i) and (ii),find expressions for the hyberbolic trigonometric functions in terms of exponential functions.

Measure the relevant physical parameters associated with the mass/spring system hanging in the front of our classroom. Set the mass into oscillation, carefully noting the initial conditions you chose. Write an expression for the subsequent motion using the “C” form to describe the sinusoidal motion.