If tanx=-3/4 and 3Π/2<x<2Π,then value of sin2x?

1 Answer
Øko
Apr 2, 2018

#sin(2x)=-24/25#

Explanation:

We seek #sin(2x)# when #tan(x)=-3/4#

By the double angle identity

#color(blue)(sin(2x)=2sin(x)cos(x)#

Let's express this in term of tangens

#sin(2x)=2sin(x)cos(x)#

#color(white)(sin(2x))=(2sin(x)cos(x))/1#

#color(white)(sin(2x))=(2sin(x)cos(x)sec^2(x))/sec^2(x)#

#color(white)(sin(2x))=(2tan(x))/sec^2(x)#

#color(white)(sin(2x))=(2tan(x))/(1+tan^2(x))#

Now let #tan(x)=-3/4#

#sin(2x)=(2(-3/4))/(1+(-3/4)^2)#

#color(white)(sin(2x))=-(3/2)/(1+9/16)#

#color(white)(sin(2x))=-(3/2*16)/(16+9)#

#color(white)(sin(2x))=-(3*8)/(25)#

#color(white)(sin(2x))=-(24)/(25)#

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