How do you find the cos of theta if $abs(sintheta)=1$ ?

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Answer 1 · 2016-05-25T06:09:54

$cos theta = 0$

No matter what the value of $theta$ , the following holds:

$cos^2 theta + sin^2 theta = 1$

Considering $sin theta$ as a Real valued function of Real numbers:

$abs(sin theta) = 1$ implies $sin theta = +-1$ , which implies $sin^2 theta = 1$

So:

$cos^2 theta = 1 - sin^2 theta = 1-1 = 0$

So:

$cos theta = 0$

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Complex footnote

The same does not hold if $theta$ can take Complex values.

For any value of $z$ we can define:

$cos z = (e^(iz)+e^(-iz))/2$

$sin z = (e^(iz)-e^(-iz))/(2i)$

For example:

If $theta = ln(1+sqrt(2))i$ then:

$sin(theta) = i$ , so $abs(sin(i)) = 1$

$cos(theta) = sqrt(2)$

How do you find the cos of theta if abs(sintheta)=1abs(sintheta)=1?