Complex Numbers Tutorial

[Isobel W]

In the complex numbers tutorial, the symbol theta is changed to phi in euler’s formula. However in the full equation for the discrete fourier transform, theta is used. What is the difference between these and why is it significant?
Thanks

[Simon]

Good – you are reading the tutorial carefully!

This is not significant – we can use any symbol we like in Euler’s formula (so long as we use the same one on the right and left-hand sides of the equation, of course!). theta and phi are popular choices for angles, in geometry.

(The Wikipedia page on Euler’s formula uses x in the text but phi in the diagram, for example).

[Vishnu M]

when doing cos(theta) = adj/hyp; or tan(theta) = opp/adj; and manipulating this to get the theta how do we know it gives us the theta we are interested and not the other angle of the triangle. for eg. how we know it gives us the one parallel to the right angle from the x-axis centre and not the one above the right angle from the x-axis centre?

[Catherine Lai]

It’s basically because the labelling of which side is opposite, adjacent or hypotenuse is defined relative to the position of the angle.

The hypotenuse is always the side opposite the right angle, with neither end touching the right angle.

The opposite side is always the side of the triangle that doesn’t touch the angle of interest, theta, at either end (and one end touches the right angle).

The adjacent is always the side of the triangle that is touches the angle of interest, theta, (but is not the hypotenuse), i.e. one end touches the right angle the other end touches the angle of interest.

Once we’ve established which angle theta in the right angled triangle that we’re interested in, the cos(theta) is defined as the length of the adjacent/hypotenuse.

In the Speech Processing course, we mainly use these trigonometric definitions to go between the polar coordinates (magnitude and angle) and rectangular (real, imaginary) representations of complex numbers, but there are some nice practical examples in this video that don’t involve complex numbers at all (just a 2 dimensional space).

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