 Now, assuming that associativity and commutativity hold for the complex numbers, we might consider their product in trigonometric form. Now, r1 and r2 are real numbers, so there's nothing special about the product r1, r2. Meanwhile, cys-theta-1, cys-theta-2, that's a more complicated expression, so let's see what happens when we multiply these together, expanding and collecting the real and imaginary parts. Now to go any further, we need to refer back to the sine or cosine of a sum, and that means we can simplify both of these expressions, and this gives us the following result. For complex numbers in trigonometric forms, the product gives us a complex number whose modulus is the product of the moduli and whose argument is the sum of the arguments. So let's try that out. Let's multiply in trigonometric form, then verify our result using the standard multiplication of complex numbers. So we need to convert both of these into polar form. Now that our numbers are in trigonometric form, they're easy to multiply. We multiply the moduli to root 2, and we add the arguments, negative pi fourth plus pi thirds. And this is a great answer, except we should always answer questions in the same language they're asked in. We're given the numbers in rectangular form, so we should answer in rectangular form. And that means we need to find the cosine of the sine of pi twelfths. Now to find these values easily, we note that they were obtained by adding negative pi fourths and pi thirds. In other words, they're pi thirds minus pi fourths, and we can use our angle sum and difference identities. Now we can check this using the standard multiplication of complex numbers, which gives us the same answer. Generally speaking, it's more trouble than it's worth to rewrite complex numbers in polar form to multiply them. An exception is when we raise a complex number to a power. Repeated applications of the multiplication formula give us the following. Assuming r, theta, r-real, and n is a whole number, r-sys-theta to the n is r-to-the-n-sys-n-theta. And this is known as Dimwav's formula, and with a name like Dimwav, you'd think that the mathematician was French. But in 1685 the government, under the influence of religious radicals, revoked a century-long policy of toleration, and Dimwav, along with many of his fellow co-religionists, led to England. So let's see how we might use the theorem of the English mathematician of Rahan-Dimwav. Let's find one plus i to the tenth power. While we could find this by expanding and using the binomial theorem, that's actually quite a bit of work, so let's use this in trigonometric form. So first we find the trigonometric form of one plus i. So Dimwav's theorem says that the power of one plus i is the modulus to the power ten, and the argument will be the argument times the power ten pi fourths. We can simplify the arithmetic to give us 32 cis 5 pi halves. But wait, remember we should always answer the question in the same language it was asked, and the complex number was given in rectangular form, so we should also answer in rectangular form. And so that means we need to find cis 5 pi halves, so computing that gives us, and we get our final answer, 32i.