What are the polar coordinates #(sqrt(7), 40.89^@)# in rectangular coordinates ?

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Answer 1 · 2017-01-11T07:25:48

#(2, sqrt(3))#

To convert polar coordinates #(r, theta)# to cartesian coordinates #(x, y)# use the formulas:

#{ (x = r cos theta), (y = r sin theta) :}#

So in our example:

#x = sqrt(7) cos 40.89^@ ~~ 2.0001025 ~~ 2.000#

#y = sqrt(7) sin 40.89^@ ~~ 1.7319324 ~~ 1.732 ~~ sqrt(3)#

Since we are given the angle to #4# significant digits, it is appropriate to round the resulting coordinates to #4# significant digits.

In case you did not recognise #1.732 ~~ sqrt(3)#, note that:

#2^2+(sqrt(3))^2 = 4+3 = 7 = (sqrt(7))^2#

satisfying Pythagoras condition for a right angled triangle.

So the point at cartesian coordinates #(2, sqrt(3))# is exactly #sqrt(7)# units from the origin.