10.3 Polar Coordinates. Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin.

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10.3 Polar Coordinates

Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin is a circle.

Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin is a circle.

One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point.

(Circle centered at the origin) (Line through the origin) Some curves are easier to describe with polar coordinates:

More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are:

Tests for Symmetry: x-axis: If (r,  ) is on the graph,so is (r, -  ).

Tests for Symmetry: y-axis: If (r,  ) is on the graph,so is (r,  -  )or (-r, -  ).

Tests for Symmetry: origin: If (r,  ) is on the graph,so is (-r,  )or (r,  +  ).

Tests for Symmetry: If a graph has two symmetries, then it has all three: 

Try graphing this on the TI-89.

To find the slope of a polar curve: We use the product rule here.

To find the slope of a polar curve:

Example:

Find the slope of the rose curve r = 2 sin 3 at the point where = π/6 and use it to find the equation of the tangent line. Finding slope of a polar curve

Find the slope of the rose curve r = 2 sin 3 at the point where = π/6 and use it to find the equation of the tangent line.

The length of an arc (in a circle) is given by r.  when  is given in radians. Area Inside a Polar Graph: For a very small , the curve could be approximated by a straight line and the area could be found using the triangle formula:

We can use this to find the area inside a polar graph.

Example: Find the area enclosed by:

Notes: To find the area between curves, subtract: Just like finding the areas between Cartesian curves, establish limits of integration where the curves cross.

Finding Area Between Curves Find the area of the region that lies inside the circle r = 1 and outside the cardioid r = 1 – cos Ø.

Finding Area Between Curves Find the area of the region that lies inside the circle r = 1 and outside the cardioid r = 1 – cos.

When finding area, negative values of r cancel out: Area of one leaf times 4:Area of four leaves:

To find the length of a curve: Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: So:

There is also a surface area equation similar to the others we are already familiar with: When rotated about the x-axis: 