Given the polar coordinates of a point, how do I find the rectangular coordinates?
(2, 3pi/4)
(-4, 7pi/6)
(2/3, -2pi/3)
Given the rectangular coordinates of a point, how do I find the polar coordinates?
(4, 0)
(3, 4)
(2, -2)
In (2,3pi/4) r = 2 and the rotation angle is 3pi/4 radians or 135º
make a sketch, draw you angle first, then go out 2 units on that ray.
drop a line to the x-axis, you now have a right-angled triangle in the II quadrant with an outside angle of 135º, and a hypotenuse of 2.
the inside angle is 45º, so it is isosceles.
Use Pythagoras to find each of the sides to be 1
But you are in quadrant II, so the point is (-1,1)
Do the others the same way
For the second type,
plot the point, then make a diagram showing the right-angled triangle.
You will have the x and the y.
Use Pythagoras to find r,
then tan ß = y/x will let you find the angle.
To convert polar coordinates to rectangular coordinates, we use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
For the first set of polar coordinates (2, 3pi/4):
x = 2 * cos(3pi/4) = -sqrt(2)
y = 2 * sin(3pi/4) = sqrt(2)
So the rectangular coordinates are (-sqrt(2), sqrt(2)).
For the second set of polar coordinates (-4, 7pi/6):
x = -4 * cos(7pi/6) = 2sqrt(3)
y = -4 * sin(7pi/6) = -2
So the rectangular coordinates are (2sqrt(3), -2).
For the third set of polar coordinates (2/3, -2pi/3):
x = (2/3) * cos(-2pi/3) = 1/3
y = (2/3) * sin(-2pi/3) = -sqrt(3)/3
So the rectangular coordinates are (1/3, -sqrt(3)/3).
To convert rectangular coordinates to polar coordinates, we use the following formulas:
r = sqrt(x^2 + y^2)
theta = arctan(y/x)
For the first set of rectangular coordinates (4, 0):
r = sqrt(4^2 + 0^2) = 4
theta = arctan(0/4) = 0
So the polar coordinates are (4, 0).
For the second set of rectangular coordinates (3, 4):
r = sqrt(3^2 + 4^2) = 5
theta = arctan(4/3) = arctan(1.33)
So the polar coordinates are (5, arctan(1.33)).
For the third set of rectangular coordinates (2, -2):
r = sqrt(2^2 + (-2)^2) = 2sqrt(2)
theta = arctan((-2)/2) = arctan(-1)
So the polar coordinates are (2sqrt(2), arctan(-1)).