Question

Find the exact value of the expression. Write the answer as a single fraction. Do not use a calculator.


`sin (3 \pi )/(2) tan ((-23 \pi )/(4)) - cos ((-10 \pi )/(3))`
Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.

Answer 1

`Use:\\sin(x+y)=sinxcosy+sinycosx\\\\tan(-x)=-tanx\\tan(k\pi+x)=tanx\\\\cos(-x)=cosx\\cos(2k\pi+x)=cosx\\cos(x+y)=cosxcosy-sinxsiny`



`sin(3\pi)/(2)=sin(\pi+(\pi)/(2))=sin\pi cos(\pi)/(2)+sin(\pi)/(2) cos\pi=0\cdot0+1\cdot(-1)=-1\\\\tan\left(-(23\pi)/(4)\right)=-tan(23\pi)/(4)=-tan\left[6\pi+\left(-(1)/(4)\pi\right)\right]=-tan\left(-(1)/(4)\pi\right)=tan(\pi)/(4)=1`


`cos\left(-(10\pi)/(3)\right)=cos(10\pi)/(3)=cos\left(2\pi+1(1)/(3)\pi\right)=cos\left(1(1)/(3)\pi\right)=cos\left(\pi+(\pi)/(3)\right)\\=cos\pi cos(\pi)/(3)-sin\pi sin(\pi)/(3)=-1\cdot(1)/(2)-0\cdot(\sqrt3)/(2)=-(1)/(2)\\\\sin(3\pi)/(2)tan\left(-(23\pi)/(4)\right)-cos\left(-(10\pi)/(3)\right)=-1\cdot1-\left(-(1)/(2)\right)=-1+(1)/(2)=-(1)/(2)`