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.

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