Find Sine, Given Tangent

Let tan x = (2ab)/(a2b2), where a > b > 0 and 0 < x < 90. Draw a right triangle with angle x and label two sides of the triangle to show tan x. Now find an expression for sin x.


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sin x = (2ab)/c.
c2  = (a2b2)2 + (2ab)2
  = a4 – 2a2b2 + b4 + 4a2b2
  = a4 + 2a2b2 + b4
  = (a2 + b2)2
So c = a2 + b2, and sin x = (2ab)/(a2 + b2)

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