What are the components of the vector between the origin and the polar coordinate $(5, pi/12)$ ?

Question

2084 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-30T22:52:00

$v_x = 5 sqrt(2+sqrt(3))/2$ ; $" "v_y= 5 sqrt(2-sqrt(3))/2$

Given: polar vector from $(0,0)$ to $(5, pi/12)$

This vector is in the first quadrant.

A polar vector: $(r, theta) = (5, pi/12)$

$x$ -component $= r cos theta = 5 cos (pi/12)$

$y$ -component $= r sin theta = 5 sin (pi/12)$

Using Half-angle formulas you can find the exact value of the angle $pi/12 = (pi/6)/2$ :

$cos (a/2) = +- sqrt((1+cos a)/2); " "sin (a/2) = +- sqrt((1-cos a)/2)$

$cos ((pi/6)/2) = sqrt((1+cos (pi/6))/2)$ ; $" "sin ((pi/6)/2) = sqrt((1-cos (pi/6))/2)$

$cos (pi/6) = cos 30^o = sqrt(3)/2$

$cos(pi/12) = sqrt((1+sqrt(3)/2)/2)$ ; $" "sin(pi/12) = sqrt((1-sqrt(3)/2)/2)$

$cos(pi/12) = sqrt(((2+sqrt(3))/2)*1/2) = sqrt(2+sqrt(3))/2$

$sin(pi/12) = sqrt(((2-sqrt(3))/2)*1/2) = sqrt(2-sqrt(3))/2$

$x$ -component $= r cos theta = 5 sqrt(2+sqrt(3))/2$

$y$ -component $=r sin theta = 5 sqrt(2-sqrt(3))/2$

What are the components of the vector between the origin and the polar coordinate (5, pi/12)(5, pi/12)?