Video: Using Trigonometry to Write Complex Numbers in Polar Form

Consider the diagram. Which of the following correctly describes the relationship between π‘Ž, π‘Ÿ, and πœƒ? [A] π‘Ž = π‘Ÿ tan πœƒ [B] π‘Ž = π‘Ÿ sin πœƒ [C] π‘Ž = (cos πœƒ)/π‘Ÿ [D] π‘Ž = π‘Ÿ cos πœƒ [E] π‘Ž = (sin πœƒ)/π‘Ÿ, Which of the following correctly describes the relationship between 𝑏, π‘Ÿ and πœƒ? [A] 𝑏 = π‘Ÿ sin πœƒ [B] 𝑏 = (cos πœƒ)/π‘Ÿ [C] 𝑏 = π‘Ÿ cos πœƒ [D] 𝑏 = (sin πœƒ)/π‘Ÿ [E] 𝑏 = π‘Ÿ tan πœƒ

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Video Transcript

Consider the diagram. Which of the following correctly describes the relationship between π‘Ž, π‘Ÿ, and πœƒ? Is it a) π‘Ž equals π‘Ÿ tan πœƒ, b) π‘Ž equals π‘Ÿ sin πœƒ, c) π‘Ž equals cos πœƒ divided by π‘Ÿ, d) π‘Ž equals π‘Ÿ cos πœƒ, or e) π‘Ž equals sin πœƒ over π‘Ÿ?

𝑧 equals π‘Ž 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 π‘Ÿ and πœƒ, where π‘Ÿ is known as the modulus of the complex number and πœƒ is the argument. Let’s consider the right-angled triangle with side lengths π‘Ž, 𝑏, and π‘Ÿ. We can use right- angled trigonometry to find an expression for π‘Ž in terms of π‘Ÿ and πœƒ.

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

And this tells us the formula that correctly describes the relationship between π‘Ž, π‘Ÿ and πœƒ. It’s d: π‘Ž equals π‘Ÿ cos πœƒ.

Which of the following correctly describes the relationship between 𝑏, π‘Ÿ and πœƒ? Is it a) 𝑏 equals π‘Ÿ sin πœƒ, b) 𝑏 equals cos πœƒ divided by π‘Ÿ, c) 𝑏 equals π‘Ÿ cos πœƒ, d) 𝑏 equals sin πœƒ over π‘Ÿ, or e) 𝑏 equals π‘Ÿ tan πœƒ?

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

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

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

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

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