# math

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if csc theta= -5/3 and theta has its terminal side in quadrant III, find the exact value of tan 2 theta.

• math -

csc(θ)=-5/3
1/sin(θ)=-5/3
sin(θ)=1/(-5/3)=-3/5
cos(θ)=-√(1-(-3/5)²)
=-4/5
tan(θ)=sin(θ)/cos(θ)
=(-3/5)/(-4/5)
=3/4
tan(2θ)=2tan(&theta))/(1-tan²(θ))
=2(3/4)/(1-(3/4)²)
=3.43
(note: 2θ is in the first quadrant.

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