Video: Using Trigonometry to Write Complex Numbers in Polar Form

Consider the diagram. Which of the following correctly describes the relationship between π, π, and π? [A] π = π tan π [B] π = π sin π [C] π = (cos π)/π [D] π = π cos π [E] π = (sin π)/π, Which of the following correctly describes the relationship between π, π and π? [A] π = π sin π [B] π = (cos π)/π [C] π = π cos π [D] π = (sin π)/π [E] π = π tan π

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Video Transcript

Consider the diagram. Which of the following correctly describes the relationship between π, π, and π? Is it a) π equals π tan π, b) π equals π sin π, c) π equals cos π divided by π, d) π equals π cos π, or e) π equals sin π over π?

π§ equals π plus ππ is known as the rectangular or sometimes algebraic form of the complex number π§. We need to find a way to represent the real and complex components of this number in terms of π and π, where π is known as the modulus of the complex number and π is the argument. Letβs consider the right-angled triangle with side lengths π, π, and π. We can use right- angled trigonometry to find an expression for π in terms of π and π.

Labelling the triangle as shown, we can see that to find an expression of π in terms of π and π, weβll need to find the trigonometric ratio that links the adjacent with the hypotenuse. Thatβs the cosine ratio; cos π is adjacent divided by hypotenuse. Substitute in what we know about the dimensions of our triangle into this formula, and we get cos π is π over π. To form an equation for π in terms of π and π, letβs multiply both sides by π. Doing so, we get π cos π is equal to π.

And this tells us the formula that correctly describes the relationship between π, π and π. Itβs d: π equals π cos π.

Which of the following correctly describes the relationship between π, π and π? Is it a) π equals π sin π, b) π equals cos π divided by π, c) π equals π cos π, d) π equals sin π over π, or e) π equals π tan π?

Letβs repeat the process from before. This time though, weβre looking to form an expression for π in terms of π and π. π is the opposite side in the triangle. Itβs a side opposite the included angle π. So we need to find a ratio that links the opposite side with hypotenuse. Thatβs the sine ratio; sin π is opposite over hypotenuse. Substituting what we know about the dimensions of our triangle into this formula, and we get sin π is equal to π over π.

Once again we can multiply both sides of this equation by π and we get π sin π is equal to π. And this means the formula that correctly describes the relationship between π, π and π is a: π equals π sin π.

Hence, express π§ in terms of π and π. We were given our complex number in rectangular or algebraic form: π§ equals π plus ππ. We now have expressions for π and π in terms of π and π. So letβs substitute these into the expression for the complex number π§. Thatβs π§ equals π cos π plus π sin π multiplied by π. We can factorize both π cos π and π sin π multiplied by π share a common factor of π.

So we can write π§ as π multiplied by cos π plus π sin π. And in doing so, we have expressed π§ in terms of π and π. You might sometimes see this called trigonometric form or polar form.