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if \(tan~x = \frac{2ab}{a^2 - b^2},\)
where a>b>0 and \(0^\circ < x < 90^\circ,\) then find sin x in terms of a and b

 Aug 5, 2020
 #1
avatar+23245 
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Picture a right triangle with angle(x) being an acute angle;

then, the side opposite to angle(x) = 2ab 

  and the side adjacent to angle(x) = a2 - b2.

 

Finding the hypotenuse:  hyp2  =  (a2 - b2)2 + (2ab)2

                                                  =  [ a4 -2a2b2 + b4 ] + [ 4a2b2 ]

                                                  =  a4 + 2a2b2 + b4 

                                                  =  (a2 + b2)2 

 

--->     hypotenuse  =  a2 + b2 

sin(x)  =  opposite / hypotenuse  =  2ab / (a2 + b2

 Aug 5, 2020

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