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Convert the equation π‘₯Β² + 𝑦² = 25 into polar form.

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Video Transcript

Convert the equation π‘₯ squared plus 𝑦 squared equals 25 into polar form.

Remember, we convert polar coordinates to Cartesian or rectangular coordinates using the formulae π‘₯ equals π‘Ÿ cos πœƒ and 𝑦 equals π‘Ÿ sin πœƒ. And these are suitable for all values of π‘Ÿ and πœƒ. In our original equation, we’ve got π‘₯ squared and 𝑦 squared. So let’s use our formulae for π‘₯ and 𝑦 to generate expressions for π‘₯ squared and 𝑦 squared in terms of π‘Ÿ and πœƒ.

Since π‘₯ is equal to π‘Ÿ cos πœƒ, it follows that π‘₯ squared must be π‘Ÿ cos πœƒ all squared, which we can distribute and say that π‘₯ squared is equal to π‘Ÿ squared times cos squared πœƒ. Similarly, we see that 𝑦 squared must be equal to π‘Ÿ sin πœƒ all squared, which is equal to π‘Ÿ squared sin squared πœƒ.

Now our original equation says that the sum of these is equal to 25. So we can say that π‘Ÿ squared cos squared πœƒ plus π‘Ÿ squared sin squared πœƒ equals 25. Our next step is to factor π‘Ÿ squared on the left-hand side of this equation. So π‘Ÿ squared times cos squared πœƒ plus sin squared πœƒ equals 25. But why did we do this?

Well, here is where it’s useful to know some of our trigonometric identities by heart. We know that cos squared πœƒ plus sin squared πœƒ is equal to one for all values of πœƒ. So we can replace cos squared πœƒ plus sin squared πœƒ in our equation with one. So π‘Ÿ squared times one equals 25. Well, we don’t need this one. π‘Ÿ squared is simply equal to 25. We solve this equation by taking the square root of both sides. And we find that π‘Ÿ is equal to five.

Remember, we would usually take both the positive and negative square root of 25. But since π‘Ÿ represents a length, we don’t need to. π‘₯ squared plus 𝑦 squared equals 25 is the same as π‘Ÿ equals five in polar form.

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