# Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an escape-time algorithm. I also see similarities to the Mandelbrot fractal in the results.

However, there's a question that's been bothering me for years. When students learn about sine and cosine, they are typically presented as functions of real numbers that describe the ratios of sides of a right triangle with a given angle as the parameter.

I am well aware that sine and cosine can be easily represented by MacLauren series, as well.

I'm also aware of trigonometry identities that relate sine and cosine to hyperbolic sine and cosine.

My question is, do sine and cosine of complex numbers have anything to do with right-triangles and circles, as sine and cosine of real numbers do?

• @حكيمالفيلسوفالضائع The question is about sine and cosine of complex numbers. – user5402 Jun 26 '14 at 22:05
• حكيم الفيلسوف الضائع linked to something about Euler's formula. Euler's formula relates the exponential of a complex number to the sine and cosine of its real and imaginary parts. My question is about the sine and cosine of complex numbers. – Vivian River Jun 26 '14 at 22:06
• I'm not sure enough to write a full answer. But I know that the hyperbolic trig functions (which arise naturally from the trig functions of complex numbers) have a lot to do with hyperbolic triangles – Mathmo123 Jun 26 '14 at 22:09
• Related Question: Do "imaginary" and "complex" angles exist? – Blue Jun 26 '14 at 22:14
• @metacompactness Ah, sorry, thought he was asking for the sin and cos in the complex plane. – Hakim Jun 26 '14 at 22:15

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

Only in part. And the other part has to do with hyperbolas. The connection is given by the fact that the equations of these two geometric shapes are of the form $x^2\pm y^2=r^2$. As you can see, there are both trigonometric as well as hyperbolic aspects to each of the two expressions below:

$$\sin(a+bi)=\sin a\cos bi+\cos a\sin bi=\sin a\cosh b+i\cos a\sinh b.$$

$$\cos(a+bi)=\cos a\cos bi-\sin a\sin bi=\cos a\cosh b-i\sin a\sinh b.$$

• Can you elaborate more about how the circle and hyperbola are connected by trig functions? – Vivian River Jul 1 '14 at 15:40
• @DanielAllenLangdon: Are you familiar with Euler's formula and the definitions of hyperbolic functions ? – Lucian Jul 1 '14 at 15:48