Find cos 𝜋 by six. Find sin 𝜋 by six. And hence, express the complex number 10 cos 𝜋 by six plus 𝑖 sin 𝜋 by six in rectangular form.
Well, 𝜋 by six radians which is 30 degrees is a special angle. And so, we remember that cos of 𝜋 by six is root three by two and sin of 𝜋 by six is a half. Alternatively, your calculator might give you these values. And now that we have these two values, we can substitute them into our complex number in trigonometric form, getting 10 times root three over two plus a half 𝑖. And distributing that 10 over the terms in parentheses, we get five root three plus five 𝑖 which as required is in rectangular form, also known as algebraic form or Cartesian form, the form 𝑎 plus 𝑏𝑖. Now, let’s convert a complex number from algebraic form to polar form.
Find the modulus of the complex number one plus 𝑖. Find the argument of the complex number one plus 𝑖. And hence, write the complex number one plus 𝑖 in polar form.
We can draw an Argand diagram to help us. And we can draw the vector from the origin zero on the Argand diagram to the complex number one plus 𝑖. The modulus of one plus 𝑖 is just the magnitude of this vector. And by considering a right triangle and applying the Pythagorean theorem, we find that this is the square root of one squared plus one squared, which is the square root of two. Of course, the formula for the modulus of 𝑎 plus 𝑏𝑖 would have given us the same answer. That’s a modulus of one plus 𝑖. What about its argument?
Well, that’s a measure of this angle here, which we’ll call 𝜃. And because we have a right-angled triangle with opposite side length one and adjacent side length one, we know that tan 𝜃 is equal to the opposite one of the adjacent one. And so, 𝜃 is arctan one over one which is arctan one which is 𝜋 by four. We also could have seen this by noticing that we’re dealing with an isosceles right triangle. And so, 𝜃 must be 45 degrees which is 𝜋 by four in radians. Now that we have the modulus and argument of our complex number, we can write it in polar form.
Our complex number is said to be written in polar form if it’s written in the form 𝑟 times cos 𝜃 plus 𝑖 sin 𝜃. And importantly for us, if the complex number 𝑧 is written in this form, then its modulus is 𝑟 and its argument is 𝜃. Well, we know the modulus and argument of our complex number so we can just substitute these in as values of 𝑟 and 𝜃. The value of 𝑟 is the modulus root two and the value of 𝜃 is the argument 𝜋 by four. This is a complex number one plus 𝑖 in polar form then. This is how we convert a number from algebraic form to polar form. We find its modulus and its argument and then we substitute these into the formula.