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    • Thread Starter

    i am happy with the idea that cos(pi radians) = cos (180 degrees) = -1. however, what about cos(3 + 4j), where j^2 = -1? i suppose it still matters whether the input to the cos function is in degrees or radians in this case, and you can see this by expanding it out. however, is it correct to say 3 + 4j radians/degrees?
    • PS Helper

    PS Helper
    When we refer to trigonometric functions with complex arguments, we require the argument to be in radians. 'Radians' doesn't make much sense with complex numbers, but I say it for the following reason.

    For z \in \mathbb{C}, we define \cos z = \displaystyle \sum_{n=0}^{\infty} \dfrac{(-)^{n+1}z^{2n}}{(2n)!} and \sin z = \displaystyle \sum_{n=0}^{\infty} \dfrac{(-)^{n+1}z^{2n+1}}{(2n+1)!} (which is in fitting with the e^{i\theta} = \cos \theta + i\sin \theta identity). This allows us to have all the usual properties of the trig functions defined over the real numbers, but extended to the whole of the complex plane. Using these definitions, we find that \cos \pi = -1 and so on; that is, the argument should be in radians, and so complex arguments correspond to the radians used when working with the reals.

    For anything beyond simple SOH-CAH-TOA stuff, radians should always be used, because the trig functions' useful properties simply don't work if we use degrees, and they don't extend to the complex plane either.
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Updated: July 28, 2010

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