How does the right angle triangle definition of sine, cosine and tangent relate to their graphs as a function of angle and to Euler's formula?

To answer this question we must understand the representation of the unit circle firstly in real coordinates and secondly in the complex plane. The unit circle is just the drawing of a circle of radius 1 in the xy-plane, represented by points (x,y) on the function x² + y² = 1. We can draw a right angle triangle with the right-hand corner lying on its circumferences, which indicates that the hypotenuse has length 1. If we wanted to find the length of the Opposite or Adjacent sides we would have to use the SOH CAH TOA mnemonic, which is the definition we use for right angle triangles: this tells us that sin(\theta) = y which is the length along the y-axis and cos(\theta) = x which is the length along the x-axis. (Sanity check: this shows us that the Pythagorean theorem a² + b² = c² holds as expected!) This works in the top-right quadrant but not in the others. If we want to work with angles beyond those between \pi/2 and 0 but all the way to +- infinity, then we extend the definition of the trig functions such that sine is the y-coord of where the triangle crosses the unit circle and cosine respectively the x-coord. Tangent follows. If you now draw a graph of sin(\theta) by tracing the y-coord as you change angle you obtain the familiar sine wave: similarly for cosine and tangent!

Now let's go one step further and draw the unit circle in the complex plane ( replace y by Im(z) and x by Re(z)). See that the value z is determined by the x-coordinate and y-coordinate and since z = x+iy, just as before where cos(\theta) = x and sin(\theta) = y we therefore have that our complex number z = cos(\theta) + i sin(\theta). By peforming the series expansion of these functions and the complex exponential e^{i \theta} we see that an arbitrary complex number z with modulus=1 can be represented by the formula e^{i \theta} = cos(\theta) + i sin(\theta). But crucially this all comes down to the unit circle!