Simplify the given expression. csc (3π /3-t)

Answer 1

Hi Bella, The problem asks us to evaluate

csc((3pi)/2-t)

which is equal to

1/sin((3pi)/2-t).

In order to simplify, we need to use the differences formula for cosine:

sin(A-B)=sinAcosB-cosAsinB.

The denominator, therefore, is equal to

sin((3pi)/2-t)=(sin((3pi)/2)cost-cos((3pi)/2)sin(t)

but cos((3pi)/2)=0 and sin((3pi)/2)=-1, so

(sin((3pi)/2)cost-cos((3pi)/2)sin(t)=(-1)(cost)-(0)(sint)=-cost.

Therefore, csc((3pi)/2-t)=1/sin((3pi)/2-t)=1/(-cost)=-sect.

I hope this helps!