Find sin2x cos2x and tan2x from the given information tanx

Find sin(2x), cos(2x), and tan(2x) from the given information. tan(x) = 12/5, x in Quadrant II sin(2x) = cos(2x) = tan(2x) = Find sin(2x), cos(2x), and tan(2x) from the given information. csc(x) = 8, tan(x)

Solution

we have given tanx=-12/5,x in 2nd quadrant

sec²x=1+tan²x

secx=sqrt(1+tan²x)=sqrt(1+144/25)=13/5

cosx=1/secx =1/(13/5)=5/13

sinx=-12/13

sin2x=2sinxcosx=2*(-12/13)*5/13 = -120/169

cos2x=2cos²x-1=2(25/169)-1=(50-169)/169=-119/169

tan2x=2tanx/(1-tan²x) =2*(-12/5)/(1-144/25)=(-24/5)/(-119/25)=(-24/5)*(25/-119)=120/119

tan2x=120/119

we have given cscx=8, tanx<0

sinx=1/cscx =1/8

cosx=sqrt(1-sin²x)=sqrt(1-1/64)=-sqrt(63)/8

tanx=sinx/cosx =(1/8)/sqrt(63)/8 =-1/sqrt(63)

sin2x=2sinxcosx=2*1/8* -sqrt(63)/8 = -sqrt(63)/32

cos2x=2cos²x-1 =2*63/64-1=(126-64)/64=62/64 =31/32

tan2x=sin2x/cos2x =sqrt(63)/32/(31/32) =-sqrt(63)/31