Question Video: Finding an Interval given the Function's Average Value over It | Nagwa Question Video: Finding an Interval given the Function's Average Value over It | Nagwa

# Question Video: Finding an Interval given the Function's Average Value over It Mathematics • Higher Education

Convert the equation π₯Β² + π¦Β² = 25 into polar form.

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### Video Transcript

Convert the equation π₯ squared plus π¦ squared equals 25 into polar form.

Remember, we convert polar coordinates to Cartesian or rectangular coordinates using the formulae π₯ equals π cos π and π¦ equals π sin π. And these are suitable for all values of π and π. In our original equation, weβve got π₯ squared and π¦ squared. So letβs use our formulae for π₯ and π¦ to generate expressions for π₯ squared and π¦ squared in terms of π and π.

Since π₯ is equal to π cos π, it follows that π₯ squared must be π cos π all squared, which we can distribute and say that π₯ squared is equal to π squared times cos squared π. Similarly, we see that π¦ squared must be equal to π sin π all squared, which is equal to π squared sin squared π.

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

Well, here is where itβs useful to know some of our trigonometric identities by heart. We know that cos squared π plus sin squared π is equal to one for all values of π. So we can replace cos squared π plus sin squared π in our equation with one. So π squared times one equals 25. Well, we donβt need this one. π squared is simply equal to 25. We solve this equation by taking the square root of both sides. And we find that π is equal to five.

Remember, we would usually take both the positive and negative square root of 25. But since π represents a length, we donβt need to. π₯ squared plus π¦ squared equals 25 is the same as π equals five in polar form.

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