Question 236884: sin (2x)=2/5
Answer by nyc_function(2741)   (Show Source):
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Sine is positive in quadrants 1 and 2.
sin2x is one of our double trig identities.
sin2x = 2sinxcosx
Given sinx = 2/5, we know that opposite side of right triangle in quadrant 2 is 2 and the hypotenuse is 5. We also know that sine = opposite/hypotenuse. We need to find the adjacent side.
Let A = adjacent side
A^2 + 2^2 = 5^2
A^2 + 4 = 25
A^2 = 25 - 4
A^2 = 21
A = sqrt{21}.
Now that we know the value of the adjacent side, we can find cosx.
cosx = adjacent side over hypotenuse.
cosx = sqrt{21}/5
Of course, cosine is negative in quadrant 2.
So, the correct answer for cosx = -sqrt{21}/5
We can now plug the values for sinx and cosx into the double trig identity for sin2x and simplify.
sin2x = 2((2/5)(-sqrt{21}/5)
sin2x = -4(sqrt{21})/25
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