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Figure 3-6 A point plotted in both polar and Cartesian
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coordinates Each radial division in the polar grid represents 1 unit Each division on the x and y axes of the Cartesian grid also represents 1 unit The shaded region is a right triangle x units wide, y units tall, and having a hypotenuse r units long
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To understand how this works, imagine what happens when r = 1 The equation r = 1 in polar coordinates gives us a unit circle We learned in Chap 2 that when we have a unit circle in the Cartesian plane, then for any point (x,y) on that circle x = cos q and y = sin q Suppose that we double the radius of the circle This makes the polar equation r = 2 The values of x and y in Cartesian coordinates both double, because when we double the length of the hypotenuse of a right triangle (such as the shaded region in Fig 3-6), we also double the lengths of the other two sides The new triangle is similar to the old one, meaning that its sides stay in the same ratio Therefore x = 2 cos q and y = 2 sin q This scheme works no matter how large or small we make the circle, as long as it stays centered at the origin If r = a, where a is some positive real number, the new right triangle is always similar to the old one, so we get x = a cos q and y = a sin q If our radius r happens to be negative, these formulas still work (For extra credit, can you figure out why )
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An example Consider the point (q,r) = (p,2) in polar coordinates Let s find the (x,y) representation of this point in Cartesian coordinates using the polar-to-Cartesian conversion formulas
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x = r cos q and y = r sin q Plugging in the numbers gives us x = 2 cos p = 2 ( 1) = 2 and y = 2 sin p = 2 0 = 0 Therefore, (x,y) = ( 2,0)
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Cartesian to polar: the radius Figure 3-6 shows us that the radius r from the origin to our point P = (x,y) is the length of the hypotenuse of a right triangle (the shaded region) that s x units wide and y units tall Using the Pythagorean theorem, we can write the formula for determining r in terms of x and y as
r = (x 2 + y 2 )1/2 That s straightforward enough Now it s time to work on the more difficult conversion: finding the polar angle for a point that s given to us in the Cartesian xy plane
The Arctangent function Before we can find the polar direction angle for a point that s given to us in Cartesian coordinates, we must be familiar with an inverse trigonometric function known as the Arctangent, which undoes the work of the tangent function (The capital A is not a typo We ll see why in a minute) Consider, for example, the fact that
tan (p /4) = 1 A true function that undoes the tangent must map an input value of 1 in the domain to an output value of p /4 in the range, but to no other values In fact, no matter what we input to the function, we must never get more than one output To ensure that the inverse of the tangent behaves as a true function, we must restrict its range (output) to an open interval where we don t get any redundancy By convention, mathematicians specify the open interval ( p /2,p /2) for this purpose When mathematicians make this sort of restriction in an inverse trigonometric function, they capitalize the first letter in the name of the function That s a code to tell us that we re working with a true function, and not a mere relation Some texts use the abbreviation tan 1 instead of Arctan to represent the inverse of the tangent function We won t use this symbol here because some readers might confuse it with the reciprocal of the tangent, which is the cotangent, not the Arctangent! If you re curious as to what the Arctangent function looks like when graphed, check out Fig 3-7 This graph consists of the principal branch of the tangent function, tipped on its side and then flipped upside-down Compare Fig 3-7 with Fig 2-5 on page 28 The principal branch of the tangent function is the one that passes through the origin Once we ve made sure we won t run into any ambiguity, we can state the above fact using the Arctangent function, getting Arctan 1 = p /4 For any real number u except odd-integer multiples of p /2 (for which the tangent function is undefined), we can always be sure that Arctan (tan u) = u Going the other way, for any real number v, we can be confident that tan (Arctan v) = v
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