What are the components of the vector between the origin and the polar coordinate (21, (-pi)/8)(21,π8)?

1 Answer
Nov 24, 2016

x = rho cos thetax=ρcosθ
y = rho sin thetay=ρsinθ

Explanation:

x = 21 cos (pi/8)x=21cos(π8)
y=-21 sin(pi/8)y=21sin(π8)

pi/8π8 is not an angle for which the value is universally known but you can find it with the trigonometric formula for the bisected angle:

cos(pi/8) = cos (1/2*pi/4) = sqrt(((1+cos(pi/4))/2))= sqrt(((1+sqrt(2))/2)cos(π8)=cos(12π4)= (1+cos(π4)2)= (1+22)
sin(pi/8) = sin (1/2*pi/4) = sqrt(((1-cos(pi/4))/2))= sqrt(((1-sqrt(2))/2)sin(π8)=sin(12π4)= (1cos(π4)2)= (122)

and eventually:

x = 21 sqrt(((1+sqrt(2))/2)x=21 (1+22)
y=-21sqrt(((1-sqrt(2))/2)y=21 (122)