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Thursday, October 4, 2012

cosθ=2

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Question: Write down one solution to each of the following:

(a) cosθ=0.5
(b) cosθ=1
(c) cosθ=0.17
(d) cosθ=2

Note that there is no ° in the question, so the answer should be in radian.

A high school student could do (a) and (b) without a calculator. Calculator will give a solution of (c) with π/2<θ<π. For (d), school algebra tells us that there is no solution, as the value of cosine lies in between -1 and 1 -- however, it is not the case if we consider complex number.

There are three definitions of cosine:

1. As the ratio of the adjacent to the hypotenuse in a right angled triangle.  Under this definition, θ can only be 0<θ<π/2 and we have 0<cosθ<1.

2. As the x-coordinate of point with polar coordinate (1,θ). Under this definition, θ can be any real number and it is still true that 1cosθ1.

3. By the formula discovered by Leonhard Euler (1707-1783):
cosθ=112!θ2+14!θ416!θ6+
or equivalently
cosθ=eiθ+eiθ2.
Nowadays, the study of wave behaviour in physics and modern axiomatic approach in mathematics, cosine is no more an "angle" thing, and therefore Euler's formula serves as a better definition.

By using Euler's formula, we can have the cosine of a complex number.
cosθ=2eiθ+eiθ2=2eiθ+eiθ=41+(eiθ)2=4eiθ(eiθ)24eiθ+1=0
One solution of the quadratic equation is eiθ=2+23 and hence one solution of (d) is θ=iloge(2+3).

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