SOLUTION: If sin x=4/5 and pi/4 < x <pi/2 , Find sin(4x).

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Question 639483: If sin x=4/5 and pi/4 < x Find sin(4x).
Answer by AnlytcPhil(1810) About Me  (Show Source):
You can put this solution on YOUR website!
sin x=4/5 and pi/4 < x < pi/2 ,
Find sin(4x)
Use the identity sin(2@) = 2sin(@)·cos(@) where @ = 2x

sin(4x) = sin[2(2x)] 

sin(4x) = 2sin(2x)·cos(2x)

Use that identity again with @=x to replace sin(2x).  Use
the identity cos(2@) = cosē(@)-sinē(@) to replace cos(2x)

sin(4x) = 2[2sin(x)·cos(x)][cosē(x)-sinē(x)]

sin(4x) = 4sin(x)·cos(x)[cosē(x)-sinē(x)] 

We have sin(x) = 4%2F5 but we don't have cos(x).  We use the identity
sinē(@)+cosē(@)=1 by solving it for cos(@)
        cosē(@)=1-sinē(@)
        cosē(x)=1-(4%2F5)ē = 1-16%2F25 = 25%2F25-16%2F25 = 9%2F25
         cos(x)=%22%22+%2B-+sqrt%289%2F25%29
         cos(x)=%22%22+%2B-+3%2F5
Sinve we are given that pi%2F4 < x < pi%2F2, we know that
the cosine is positive, therefore
         cos(x)=3%2F5

sin(4x) = 4sin(x)·cos(x)[cosē(x)-sinē(x)]
sin(4x) = 4(4%2F5)·3%2F5[%283%2F5%29%5E2-%284%2F5%29%5E2]
sin(4x) = 48%2F25[9%2F25-16%2F25]
sin(4x) = 48%2F25[-7%2F25]
sin(4x) = -336%2F625

Edwin