MHB How Can You Determine the Values of ab+cd Given These Equations?

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    2016
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The discussion focuses on evaluating the expression ab + cd given the equations a² + b² = 4, c² + d² = 4, and ac + bd = 2. Participants explore the implications of these equations on the values of a, b, c, and d. The correct solutions provided by members highlight the relationships between the variables and their geometric interpretations. The final values for ab + cd are derived through algebraic manipulation and analysis of the constraints. The thread emphasizes problem-solving techniques in mathematics.
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Here is this week's POTW:

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Suppose that 4 real numbers $a,\,b,\,c,\,d$ satisfy the conditions as shown below:

$a^2+b^2=4$
$c^2+d^2=4$
$ac+bd=2$

Evaluate all possible values for $ab+cd$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to the following members for their correct solution::)

1. greg1313
2. kaliprasad

Solution from greg1313:
Let $a=2\cos(x),b=2\sin(x),c=2\cos(y),d=2\sin(y)$ then the first two conditions are satisfied.

For $ac+bd$ we have $4\cos(x)\cos(y)+4\sin(x)\sin(y)=4\cos(x-y)=2$ so we must have $x-y=\pm\dfrac{\pi}{3}+2k\pi,k\in\mathbb Z$

From all of that, $ab+cd=4\cos(x)\sin(x)+4\cos(y)\sin(y)=2(\sin(2x)+\sin(2y))=2(2\sin(x+y)\cos(x-y))=2\sin(x+y)$

Hence $-2\le ab+cd\le2$
 
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