Algebra.Com's Answer #648394 by ikleyn(52776)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Convert the polar equation to cartesian/rectangular coordinates\r
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document.write( "(r^2) cos (2 theta) = 2 \r
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document.write( "Thank you!
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document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
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document.write( "Let's do it together.\r\n" );
document.write( "We start with the point ( , ) in a coordinate plane. This point is presented in the polar form, \r\n" );
document.write( "and we want to learn what is a curve  =  .\r\n" );
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document.write( "Do you know that  =  ?\r\n" );
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document.write( "It is the direct consequence of the addition formula for cosine.\r\n" );
document.write( "Also, it is named the formula for double argument for cosine.\r\n" );
document.write( "If you don't know it, then look in these lessons \r\n" );
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document.write( " Addition and subtraction formulas\r\n" );
document.write( " Trigonometric functions of multiply argument\r\n" );
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document.write( "OK. Having this, let's make one step further.\r\n" );
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document.write( " =  =  . =  . . (1)\r\n" );
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document.write( "Now, is x-component of the point (, ) in the rectangular coordinate system on the coordinate plane. \r\n" );
document.write( "So we can write  = .\r\n" );
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document.write( "Similarly, is y-component of the point (,) in the rectangular coordinate system on the coordinate plane. \r\n" );
document.write( "So we can write  = .\r\n" );
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document.write( "Therefore, we can rewrite the formula (1) in this way:\r\n" );
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document.write( " = . =  =  . (3)\r\n" );
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document.write( "Then your original equation  = becomes \r\n" );
document.write( " = .\r\n" );
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document.write( "Do you recognize this equation ? What is this ? Is it familiar to you ?\r\n" );
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document.write( "But of course, it is the equation of a hyperbola . . . \r\n" );
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document.write( "So, we unraveled this mystery.\r\n" );
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document.write( "Comment from student: It looks like the equation for a circle to me. A circle with a diameter of 2 centered at (0,0), correct? Thank you for your help!
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document.write( "My response: Do not make a mistake !\r
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document.write( " It is definitely the equation of a HYPERBOLA.\r
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document.write( "About equations for hyperbola see the lesson Hyperbola definition, canonical equation, characteristic points and elements in this site.\r
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