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Zenmo
 one year ago
Question on Trig. Identities on this problem that involves converting rectangular equation to polar form for xy=16.
Zenmo
 one year ago
Question on Trig. Identities on this problem that involves converting rectangular equation to polar form for xy=16.

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Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0\[xy=16 (1st step), (rcos \theta)(rsin \theta)=16 (2nd step), (\cos \theta)(\sin \theta)r^2=16 (3rd step), \frac{ 1 }{ 2 }(\sin2 \theta)r^2=16 (4th step).\] How do I get from the 3rd step to the 4th step?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0x = r cos Θ y = r sin Θ xy = 16 (r sin Θ)(r cos Θ) = 16 r² sin Θ cos Θ = 16

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1From the third step to the 4th? You apply your Sine Double Angle Identity :)

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[\Large\rm \color{orangered}{2\sin \theta \cos \theta=\sin(2\theta)}\]We want to make that show up in our problem. When we're only given this:\[\Large\rm \sin \theta \cos \theta\]We'll make a 2 show up by double two things at once, multiplying by 2, and dividing by 2.\[\Large\rm \frac{1}{2}\cdot2\cdot\sin \theta \cos \theta\]And we can apply our identity from there :)\[\Large\rm \frac{1}{2}\cdot\color{orangered}{2\sin \theta \cos \theta}\]

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1by doing* two things at once, blah typo

Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0\[(\cos \theta)(\sin \theta)r^2=16\] Could u do the next step for that?

Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0Just a tiny bit confused on the double angle identity part

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1\[(\cos \theta)(\sin \theta)r^2=16\]\[\sin \theta \cos \theta r^2=16\]\[\frac{1}{2}\cdot2\cdot \sin \theta \cos \theta r^2=16\]\[\frac{1}{2}\cdot\color{orangered}{2\cdot \sin \theta \cos \theta} r^2=16\]\[\frac{1}{2}\cdot\color{orangered}{\sin (2 \theta)} r^2=16\]

Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0Why do we multiply by 2 and divide by 2 for the 2? That is the only part that I don't get, other than that, I understand on how to do rest of the problem.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Well you're probably used to this simple trick in math, do something to one side, do it to the other as well. We're using a different trick, do something to one side, undo it at the same time. Example: \(\Large\rm x= x2+2\) Maybe I really need an x2 to show up so I can use it in some way. Like in this example:\[\Large\rm \frac{x}{x2}\]Yes, we could go through the process of long division, but instead we could add and subtract 2 in the numerator,\[\Large\rm =\frac{x2+2}{x2}=\frac{x2}{x2}+\frac{2}{x2}=1+\frac{2}{x2}\]And that's how we would divide the example I gave.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1We're multiplying by 2, we're also dividing by 2 because we have to keep things balanced. Then we `use` the 2 which is multiplying to fix our identity.

Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0Ok, thanks, I got it now after looking at your examples

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1You can think of it like this if it helps maybe,\[\Large\rm \sin \theta \cos \theta=\frac{2}{2}\sin \theta \cos \theta=\frac{1}{2}\cdot 2\sin \theta \cos \theta\]Ok cool :)

Zenmo
 one year ago
Best ResponseYou've already chosen the best response.0yea, that definitely helps by thinking it as "keeping the balance." :D
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