What are the components of the vector between the origin and the polar coordinate (-2, (5pi)/8)?

1 Answer

x=sqrt(2-sqrt2)=0.765367 and
y=-sqrt(2+sqrt2)=-1.84776

Explanation:

the x component

x=r cos theta
x=-2*cos ((5pi)/8)

but cos ((5pi)/8)=-sin(pi/8)=-sqrt((1-cos(pi/4))/2)=-sqrt((1-1/sqrt2)/2)

cos ((5pi)/8)=-1/2sqrt(2-sqrt2)

so that

x=-2(-1/2sqrt(2-sqrt2))

x=sqrt(2-sqrt2)=0.765367

the y component

y=r sin theta

y=-2 sin ((5pi)/8)

but sin ((5pi)/8)=cos (pi/8)

cos (pi/8)=sqrt((1+cos(pi/4))/2)=sqrt((1+1/sqrt2)/2)=1/2sqrt(2+sqrt2)

and

y=-2*(1/2sqrt(2+sqrt2))

y=-sqrt(2+sqrt2)=-1.84776

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