decorative picture of space events
Aeronautics Home PageVirtual Visits (Video Conferencing, Virtual Tours, Webcasts, plus more.)Math and Science ResourcesTeachers ResourcesInternet Access Research ArchiveLink to Learning Technologies Project (LTP) Homepage

       Math & Science Home | Proficiency Tests | Mathematical Thinking in Physics | Aeronauts 2000



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

Motivational Argument for the Expression: eix = cos x + i sin x

Problem: Show that eix = cos x + i sin x, where i = square root of negative one

Solution: Let us turn to the theory of differential equations. The equation

y'' + y = 0

(where the prime notation symbolizes differentiation with respect to x) has a solution of the form y = alphacos x + betasin x. This same differential equation also has a solution of the form y =yeix + differentiale-ix. [This latter form is obtained by assuming the ansatz, y = em, and using it in the original equation to obtain the condition on m that m2 + 1 = 0, from which m = ± i.]

We may now equate

alphacos x + betasin x = yeix + differentiale-ix (eq. 1)

We wish to solve for eix. Differentiating once and multiplying the result by -i gives

(- i) [(-alphasin x) + (betacos x)] = yeix -differentiale-ix
i [alphasin x - betacos x] = yeix -differentiale-ix (eq.2)

Adding eqs. 1 and 2 gives

2yeix= (alpha- ibeta) cos x + (beta + ialpha) sin x (eq. 3)
eix = (alpha* - ibeta*) cos x + (beta* + ialpha*) sin x (eq. 3a)

where alpha* = alpha/2y and beta* = beta/2y. We must now determine values for alpha* and beta*. We do so by setting x = 0 and observing that ei0 = e0 = 1. Also, cos 0 = 1 and sin 0 = 0. In this case, eq. 3a reduces to

1 = alpha* - ibeta*

Equating the real and imaginary parts immediately gives

alpha* = 1, and beta* = 0.

With these values, eq. 3a yields the required expression

eix = cos x + i sin x Q.E.D.

Please send suggestions/corrections to:
Web Related:
Technology Related:
Responsible NASA Official: Theresa.M.Scott (Acting)