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Figure 3-7 A graph of the Arctangent function The
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domain extends over all the real numbers The range is restricted to values larger than p /2 and smaller than p /2 Each division on the y axis represents p /6 units
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Cartesian to polar: the angle We now have the tools that we need to determine the polar angle q for a point on the basis of its Cartesian coordinates x and y We already know that
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x = r cos q and y = r sin q As long as x 0, it follows that y /x = (r sin q)/(r cos q) = (r /r)(sin q)/(cos q) = (sin q)/(cos q) = tan q Simplifying, we get tan q = y /x
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If we take the Arctangent of both sides, we obtain Arctan (tan q ) = Arctan ( y /x) which can be rewritten as q = Arctan (y /x) Suppose the point P = (x,y) happens to lie in the first or fourth quadrant of the Cartesian plane In this case, we have p /2 < q < p /2 so we can directly use the conversion formula q = Arctan ( y /x) If P = (x,y) is in the second or third quadrant, then we have p /2 < q < 3p /2 That s outside the range of the Arctangent function, but we can remedy this situation if we subtract p from q When we do this, we bring q into the allowed range but we don t change its tangent, because the tangent function repeats itself every p radians (If you look back at Fig 2-5 again, you will notice that all of the branches in the graph are identical, and any two adjacent branches are p radians apart) In this situation, we have q p = Arctan ( y /x) which can be rewritten as q = p + Arctan ( y /x) Now we re ready to derive specific formulas for q in terms of x and y Let s break the scenario down into all possible general locations for P = (x,y), and see what we get for q in each case: P at the origin If x = 0 and y = 0, then q is theoretically undefined However, let s assign q a default value of 0 at the origin By doing that, we can fill the hole that would otherwise exist in our conversion scheme P on the +x axis If x > 0 and y = 0, then we re on the positive x axis We can see from Fig 3-6 that q = 0 P in the first quadrant If x > 0 and y > 0, then we re in the first quadrant of the Cartesian plane where q is larger than 0 but less than p /2 We can therefore directly apply the conversion formula q = Arctan ( y /x)
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Polar Two-Space
P on the +y axis If x = 0 and y > 0, then we re on the positive y axis We can see from Fig 3-6 that q = p /2 P in the second quadrant If x < 0 and y > 0, then we re in the second quadrant of the Cartesian plane where q is larger than p /2 but less than p In this case, we must apply the modified conversion formula q = p + Arctan ( y /x) P on the -x axis If x < 0 and y = 0, then we re on the negative x axis We can see from Fig 3-6 that q = p P in the third quadrant If x < 0 and y < 0, then we re in the third quadrant of the Cartesian plane where q is larger than p but less than 3p /2, so we apply the modified conversion formula q = p + Arctan ( y /x) P on the -y axis If x = 0 and y < 0, then we re on the negative y axis We can see from Fig 3-6 that q = 3p /2 P in the fourth quadrant If x > 0 and y < 0, then we re in the fourth quadrant of the Cartesian plane where q is larger than 3p /2 but smaller than 2p That s the same thing as saying that p /2 < q < 0 We ll get an angle in that range if we apply the original conversion formula q = Arctan ( y /x) In the interest of elegance, we d like the angle in the polar representation of a point to always be nonnegative but less than 2p We can make this happen by adding in a complete rotation of 2p to the basic conversion formula, getting q = 2p + Arctan ( y /x) We have taken care of all the possible locations for P A summary of the nine-part conversion formula that we ve developed is given in the following table
q=0 q=0 q = Arctan ( y /x) q = p /2 q = p + Arctan ( y /x) q=p q = p + Arctan ( y /x) q = 3p /2 q = 2p + Arctan ( y /x)
At the origin On the +x axis In the first quadrant On the +y axis In the second quadrant On the x axis In the third quadrant On the y axis In the fourth quadrant
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