document.write( "Question 395841: Hi, I have a big test on complex numbers tomorrow and I am having a very hard time with this particular question:\r
\n" ); document.write( "\n" ); document.write( "Directions: On a complex plane, a point z has been graphed. your\r
\n" ); document.write( "\n" ); document.write( "If cosθ=(-8)/(22) and (π)/(2)≤θ≤π:\r
\n" ); document.write( "\n" ); document.write( "a) Find z in standard form.
\n" ); document.write( "b) Find z in trigonometric form. (Use whole degrees)\r
\n" ); document.write( "\n" ); document.write( "θ= Theta
\n" ); document.write( "π= pi \n" ); document.write( "

Algebra.Com's Answer #280946 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
Hi, I have a big test on complex numbers tomorrow and I am having a very hard time with this particular question:
\n" ); document.write( "Directions: On a complex plane, a point z has been graphed.
\n" ); document.write( "If cosθ=(-8)/(22) and (π)/(2)≤θ≤π:
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document.write( "The other tutor messed up, although he's right about the 2nd quadrant\r\n" );
document.write( "between  and .\r\n" );
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document.write( "Here is the point:\r\n" );
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document.write( "We draw a line from the origin to the point.  The length of that line is\r\n" );
document.write( "indicated by the letter \"r\"\r\n" );
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document.write( "Now we indicate θ with a red arc that starts on the right\r\n" );
document.write( "side of the x-axis and swings counter-clockwise to the green\r\n" );
document.write( "line:\r\n" );
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document.write( "Next we draw a perpendicular from the point to the x-axis. The blue\r\n" );
document.write( "line below, which we label as y.\r\n" );
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document.write( "Now we have a right triangle, with the horizontal leg labeled as x,\r\n" );
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document.write( "Since we are given = cos(θ) =  and we know that\r\n" );
document.write( "cos(θ) = , we will take x to be -8 and r to be 22,\r\n" );
document.write( "So we label those:\r\n" );
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document.write( "Now we calculate y by the Pythagorean theorem:\r\n" );
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document.write( "     r² = x² + y²\r\n" );
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document.write( "  (22)² = (-8)² + y²\r\n" );
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document.write( "    484 = 64 + y²\r\n" );
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document.write( "    420 = y²\r\n" );
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document.write( "   ⎷420 = y\r\n" );
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document.write( " ⎷4*105 = y\r\n" );
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document.write( "  2⎷105 = y\r\n" );
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document.write( "So we label y:\r\n" );
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document.write( "Therefore the given point, call it P has the coordinates P(-8, 2⎷105).\r\n" );
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document.write( "Therefore z in standard form is x + yi = -8 + 2⎷105*i \r\n" );
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document.write( "z in trigonometric form is r(cosθ + isinθ)\r\n" );
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document.write( "But we must find θ by using the given cosθ = \r\n" );
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document.write( "We use the inverse cosine of the POSITIVE  to find the\r\n" );
document.write( "reference angle of θ to be 68.7° or as a whole number of degrees,\r\n" );
document.write( "69°.  But to get that in the second quadrant we subtract from 180°\r\n" );
document.write( "and get θ = 111°\r\n" );
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document.write( "Now the trig form\r\n" );
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document.write( "r(cosθ + isinθ)\r\n" );
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document.write( "becomes \r\n" );
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document.write( "2⎷105(cos111° + i*sin111°)\r\n" );
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document.write( "Edwin

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