Question Video: Converting Complex Numbers from Trigonometric to Rectangular Form Mathematics • Third Year of Secondary School

Video Transcript

Find cos of 𝜋 by six. Find sin of 𝜋 by six. Hence, express the number 10 cos 𝜋 by six plus 𝑖 sin 𝜋 by six in rectangular form.

There are three bits to this question. The first two parts will allow us to answer the third part. So let’s begin with part one, and we’ll evaluate cos of 𝜋 by six. Now, of course, we could use a calculator to do so. But this is one of the exact values we should know by heart. Let’s draw the table that helps us calculate really quickly the exact values of cos, sin, and tan of zero, 𝜋 by six, 𝜋 by four, 𝜋 by three, and 𝜋 by two.

A quick way to remember this is as shown. We begin with zero, one, two, three, and four. And then we repeat this but in the opposite direction. Then, the denominator of every single expression is going to be equal to two. Finally, we take the square root of the numerator of every single one of these expressions. But this allows us to simplify quite quickly some of these. The square root of zero divided by two is zero. Then, we know that the square root of one is just one. So sin of 𝜋 by six and cos of 𝜋 by three is one-half. We also know that the square root of four is two. And then we get two divided by two, which is one. So sin of 𝜋 by two and cos of zero is equal to one.

So we have our values for sine and cosine. But by remembering that tan is sin divided by cos, we can also fill out the third row in this table. We take each individual value for sin and divide it by each individual value for cos. When we have the fractions, we can achieve this by simply looking at the numerators since the denominators are equal. tan of zero is zero divided by one, which is zero. Then, tan of 𝜋 by six is one over root three. tan of 𝜋 by four is root two over root two, which is one. tan of 𝜋 by three is root three over one, which is just root three. Then, tan of 𝜋 by two is one divided by zero, which is of course undefined.

So now that we have the relevant values for sine, cosine, and tan, we can answer the question. First, we’re interested in cos of 𝜋 by six, so that must be root three over two. Next, we’re looking to find the value of sin of once again 𝜋 by six. So that’s one-half. Next, we’re told to express the number 10 cos 𝜋 by six plus 𝑖 sin 𝜋 by six in rectangular form. Well, that’s simply of the form 𝑎 plus 𝑖𝑏, where 𝑎 and 𝑏 are real numbers. They are, respectively, called the real and imaginary parts of the complex number.

By replacing cos of 𝜋 by six with root three over two and sin of 𝜋 by six with one-half, this number can be written as 10 times root three over two plus 𝑖 times one-half. Now, in fact, we can distribute the 10 over the parentheses, which essentially results in multiplying root three plus one 𝑖 by five. That gives us five root three plus five 𝑖. And so we have our complex number in rectangular form. It’s five root three plus five 𝑖.