# Video: Using Trigonometry to Write Complex Numbers in Polar Form

Consider the diagram. Which of the following correctly describes the relationship between 𝑎, 𝑟, and 𝜃? [A] 𝑎 = 𝑟 tan 𝜃 [B] 𝑎 = 𝑟 sin 𝜃 [C] 𝑎 = (cos 𝜃)/𝑟 [D] 𝑎 = 𝑟 cos 𝜃 [E] 𝑎 = (sin 𝜃)/𝑟, Which of the following correctly describes the relationship between 𝑏, 𝑟 and 𝜃? [A] 𝑏 = 𝑟 sin 𝜃 [B] 𝑏 = (cos 𝜃)/𝑟 [C] 𝑏 = 𝑟 cos 𝜃 [D] 𝑏 = (sin 𝜃)/𝑟 [E] 𝑏 = 𝑟 tan 𝜃

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### Video Transcript

Consider the diagram. Which of the following correctly describes the relationship between 𝑎, 𝑟, and 𝜃? Is it a) 𝑎 equals 𝑟 tan 𝜃, b) 𝑎 equals 𝑟 sin 𝜃, c) 𝑎 equals cos 𝜃 divided by 𝑟, d) 𝑎 equals 𝑟 cos 𝜃, or e) 𝑎 equals sin 𝜃 over 𝑟?

𝑧 equals 𝑎 plus 𝑏𝑖 is known as the rectangular or sometimes algebraic form of the complex number 𝑧. We need to find a way to represent the real and complex components of this number in terms of 𝑟 and 𝜃, where 𝑟 is known as the modulus of the complex number and 𝜃 is the argument. Let’s consider the right-angled triangle with side lengths 𝑎, 𝑏, and 𝑟. We can use right- angled trigonometry to find an expression for 𝑎 in terms of 𝑟 and 𝜃.

Labelling the triangle as shown, we can see that to find an expression of 𝑎 in terms of 𝑟 and 𝜃, we’ll need to find the trigonometric ratio that links the adjacent with the hypotenuse. That’s the cosine ratio; cos 𝜃 is adjacent divided by hypotenuse. Substitute in what we know about the dimensions of our triangle into this formula, and we get cos 𝜃 is 𝑎 over 𝑟. To form an equation for 𝑎 in terms of 𝑟 and 𝜃, let’s multiply both sides by 𝑟. Doing so, we get 𝑟 cos 𝜃 is equal to 𝑎.

And this tells us the formula that correctly describes the relationship between 𝑎, 𝑟 and 𝜃. It’s d: 𝑎 equals 𝑟 cos 𝜃.

Which of the following correctly describes the relationship between 𝑏, 𝑟 and 𝜃? Is it a) 𝑏 equals 𝑟 sin 𝜃, b) 𝑏 equals cos 𝜃 divided by 𝑟, c) 𝑏 equals 𝑟 cos 𝜃, d) 𝑏 equals sin 𝜃 over 𝑟, or e) 𝑏 equals 𝑟 tan 𝜃?

Let’s repeat the process from before. This time though, we’re looking to form an expression for 𝑏 in terms of 𝑟 and 𝜃. 𝑏 is the opposite side in the triangle. It’s a side opposite the included angle 𝜃. So we need to find a ratio that links the opposite side with hypotenuse. That’s the sine ratio; sin 𝜃 is opposite over hypotenuse. Substituting what we know about the dimensions of our triangle into this formula, and we get sin 𝜃 is equal to 𝑏 over 𝑟.

Once again we can multiply both sides of this equation by 𝑟 and we get 𝑟 sin 𝜃 is equal to 𝑏. And this means the formula that correctly describes the relationship between 𝑏, 𝑟 and 𝜃 is a: 𝑏 equals 𝑟 sin 𝜃.

Hence, express 𝑧 in terms of 𝑟 and 𝜃. We were given our complex number in rectangular or algebraic form: 𝑧 equals 𝑎 plus 𝑏𝑖. We now have expressions for 𝑎 and 𝑏 in terms of 𝑟 and 𝜃. So let’s substitute these into the expression for the complex number 𝑧. That’s 𝑧 equals 𝑟 cos 𝜃 plus 𝑟 sin 𝜃 multiplied by 𝑖. We can factorize both 𝑟 cos 𝜃 and 𝑟 sin 𝜃 multiplied by 𝑖 share a common factor of 𝑟.

So we can write 𝑧 as 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃. And in doing so, we have expressed 𝑧 in terms of 𝑟 and 𝜃. You might sometimes see this called trigonometric form or polar form.