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

May 25, 2016

$\cos \theta = 0$

#### Explanation:

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:

$\left\mid \sin \theta \right\mid = 1$ implies $\sin \theta = \pm 1$, 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:

$\cos z = \frac{{e}^{i z} + {e}^{- i z}}{2}$

$\sin z = \frac{{e}^{i z} - {e}^{- i z}}{2 i}$

For example:

If $\theta = \ln \left(1 + \sqrt{2}\right) i$ then:

$\sin \left(\theta\right) = i$, so $\left\mid \sin \left(i\right) \right\mid = 1$

$\cos \left(\theta\right) = \sqrt{2}$