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 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... Sunlight Exerts Pressure Falling Eastward What if an Asteroid Hit the Earth Using a Jeep to Estimate the Energy in Gasoline How do Police Radars really work? How "Fast" is the Speed of Light? How Long is a Light Year? How Big is a Trillion? "Seeing" the Earth, Moon, and Sun to Scale Of Stars and Drops of Water If I Were to Build a Model of the Cosmos... A Number Trick Designing a High Altitude Balloon Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water Calendar Calculations Telling Time by the Stars - Sidereal Time Fields, an Heuristic Approach The Irrationality of The Irrationality of The Number (i)i Estimating the Temperature of a Flat Plate in Low Earth Orbit Proving that (p)1/n is Irrational when p is a Prime and n>1 The Transcendentality of Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions Maxwell's Equations: The Vector and Scalar Potentials A Possible Scalar Term Describing Energy Density in the Gravitational Field A Proposed Relativistic, Thermodynamic Four-Vector Motivational Argument for the Expression-eix=cosx+isinx Another Motivational Argument for the Expression-eix=cosx+isinx Calculating the Energy from Sunlight over a 12 hour period Calculating the Energy from Sunlight over actual full day Perfect Numbers-A Case Study Gravitation Inside a Uniform Hollow Sphere Further note on Gravitation Inside a Uniform Hollow Sphere Pythagorean Triples Black Holes and Point Set Topology Additional Notes on Black Holes and Point Set Topology Field Equations and Equations of Motion (General Relativity) The observer in modern physics A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File

##### Another Motivational Argument for the Expression: eiq= cos q+ i sin q

Problem: Show that eiq = cos q+ i sin q, where i = .

Solution: Let z = x + iy be any complex number. We know, from geometry, that

z = x + iy = r(cos q + i sin q).

In the previous article on eix, we used the theory of differential equations to establish the required identity. This time, we will use the natural logarithm function ln(z) to establish that same identity.
Let us form the function ln(z):

ln(z) = ln(x + iy)
= ln [r(cos q + i sin q)]
= ln( ) + ln(cos q + i sin q).

The first term in the third line, ln( ), involves the real number , and so will concern us no further here. The second term, ln(cos q + i sin q), involves a complex number whose magnitude is unity.
Let us set

u(q) = ln(cos q + i sin q)

Then

eu(q) = eln(cos q + i sin q) = cos q + i sin q.

Our problem thus reduces to showing that u(q) = iq .We notice immediately that

eu(0) = cos 0 + i sin 0 = 1

which gives us the identity

u(0) = 1.

We now differentiate the expression eu = cos q + i sin q to obtain

deu = eu du = (- sin q + i cos q) dq
or
eu = (- sin q + i cos q) dq/du = cos q + i sin q.

We now have the derivative dq/du:

dq/du = (cos q + i sin q)/(- sin q + i cos q)
or
du = [(- sin q + i cos q)/(cos q + i sin q)] dq

Multiplying by unity in the form (cos q - i sin q)/(cos q - i sin q) allows us to simplify the right-hand side, giving

du = i dq.

Integrating, we acquire

u = iq + C.

But since we already know that u = 0 when q = 0, we have that the constant C = 0. Therefore

u = iq,

which establishes the required identity.