Example 1. Example 1 Plot the points whose polar coordinates are given by


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1 Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points in the xyplane, using the origin (0, 0) and the positive xaxis for reference. A point P in the plane, has polar coordinates (r, θ), where r is the distance of the point from the origin and θ is the angle that the ray OP makes with the positive xaxis.
2 Example 1 Example 1 Plot the points whose polar coordinates are given by (, ) (3, ) (3, 7 ) (, 5 )
3 Example 1 Example 1 Plot the points whose polar coordinates are given by (, ) (3, ) (3, 7 ) (, 5 )
4 Example 1 Note the representation of a point in polar coordinates is not unique. For instance for any θ the point (0, θ) represents the pole O. We extend the meaning of polar coordinate to the case when r is negative by agreeing that the two points (r, θ) and ( r, θ) are in the same line through O and at the same distance r but on opposite side of O. Thus Example Plot the point ( 3, 3 ) ( r, θ) = (r, θ + )
5 Example 1 Note the representation of a point in polar coordinates is not unique. For instance for any θ the point (0, θ) represents the pole O. We extend the meaning of polar coordinate to the case when r is negative by agreeing that the two points (r, θ) and ( r, θ) are in the same line through O and at the same distance r but on opposite side of O. Thus Example Plot the point ( 3, 3 ) ( r, θ) = (r, θ + )
6 Polar to Cartesian coordinates To convert from Polar to Cartesian coordinates, we use the identities: x = r cos θ, y = r sin θ Example 3 Convert the following (given in polar coordinates) to Cartesian coordinates (, ) and (3, 3 )
7 Polar to Cartesian coordinates To convert from Polar to Cartesian coordinates, we use the identities: x = r cos θ, y = r sin θ Example 3 Convert the following (given in polar coordinates) to Cartesian coordinates (, ) and (3, 3 ) For (, ), we have r =, θ =. In Cartesian coordinates, we get x = r cos θ = cos(/) = 1 = we get y = r sin θ = sin(/) = 1 =
8 Polar to Cartesian coordinates To convert from Polar to Cartesian coordinates, we use the identities: x = r cos θ, y = r sin θ Example 3 Convert the following (given in polar coordinates) to Cartesian coordinates (, ) and (3, 3 ) For (, ), we have r =, θ =. In Cartesian coordinates, we get x = r cos θ = cos(/) = 1 = we get y = r sin θ = sin(/) = 1 = For (3, ), we have r = 3, θ =. In Cartesian coordinates, we get 3 3 x = r cos θ = 3 cos( ) = 3 1 = 3 3 we get y = r sin θ = 3 sin( ) = 3 3 = 3 3 3
9 Cartesian to Polar coordinates To convert from Cartesian to polar coordinates, we use the following identities r = x + y, tan θ = y x When choosing the value of θ, we must be careful to consider which quadrant the point is in, since for any given number a, there are two angles with tan θ = a, in the interval 0 θ. Example 3 Give polar coordinates for the points (given in Cartesian coordinates) (, ), (1, 3), and ( 1, 3)
10 Cartesian to Polar coordinates To convert from Cartesian to polar coordinates, we use the following identities r = x + y, tan θ = y x When choosing the value of θ, we must be careful to consider which quadrant the point is in, since for any given number a, there are two angles with tan θ = a, in the interval 0 θ. Example 3 Give polar coordinates for the points (given in Cartesian coordinates) (, ), (1, 3), and ( 1, 3) For (, ), we have x =, y =. Therefore r = x + y = + = 8, and r = 8. We have tan θ = y = / = 1. x Since this point is in the first quadrant, we have θ =. Therefore the polar coordinates for the point (, ) are ( 8, ).
11 Cartesian to Polar coordinates To convert from Cartesian to polar coordinates, we use the following identities r = x + y, tan θ = y x When choosing the value of θ, we must be careful to consider which quadrant the point is in, since for any given number a, there are two angles with tan θ = a, in the interval 0 θ. Example 3 Give polar coordinates for the points (given in Cartesian coordinates) (, ), (1, 3), and ( 1, 3) For (, ), we have x =, y =. Therefore r = x + y = + = 8, and r = 8. We have tan θ = y = / = 1. x Since this point is in the first quadrant, we have θ =. Therefore the polar coordinates for the point (, ) are ( 8, ). For (1, 3), we have x = 1, y = 3. Therefore r = x + y = =, and r =. We have tan θ = y = 3. x Since this point is in the fourth quadrant, we have θ =. 3 Therefore the polar coordinates for the point (1, 3) are (, ). 3
12 Example 3 Example 3 Give polar coordinates for the point (given in Cartesian coordinates) ( 1, 3)
13 Example 3 Example 3 Give polar coordinates for the point (given in Cartesian coordinates) ( 1, 3) For ( 1, 3), we have x = 1, y = 3. Therefore r = x + y = =, and r =. We have tan θ = y x = 3. Since this point is in the second quadrant, we have θ = 3. Therefore the polar coordinates for the point ( 1, 3) are (, 3 ).
14 Graphing Equations in Polar Coordinates The graph of an equation in polar coordinates r = f (θ) or F (r, θ) = 0 consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Example Graph the following equations r = 5, θ =
15 Graphing Equations in Polar Coordinates The graph of an equation in polar coordinates r = f (θ) or F (r, θ) = 0 consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Lines: A line through the origin (0, 0) has equation θ = θ 0 Example Graph the following equations r = 5, θ =
16 Graphing Equations in Polar Coordinates The graph of an equation in polar coordinates r = f (θ) or F (r, θ) = 0 consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Lines: A line through the origin (0, 0) has equation θ = θ 0 Circle centered at the origin: A circle of radius r 0 centered at the origin has equation r = r 0 in polar coordinates. Example Graph the following equations r = 5, θ =
17 Graphing Equations in Polar Coordinates The graph of an equation in polar coordinates r = f (θ) or F (r, θ) = 0 consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Lines: A line through the origin (0, 0) has equation θ = θ 0 Circle centered at the origin: A circle of radius r 0 centered at the origin has equation r = r 0 in polar coordinates. Example Graph the following equations r = 5, θ = The equation r = 5 describes a circle of radius 5 centered at the origin. The equation θ = describes a line through the origin making an angle with the positive x axis. of
18 Graphing Equations in Polar Coordinates The graph of an equation in polar coordinates r = f (θ) or F (r, θ) = 0 consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. Lines: A line through the origin (0, 0) has equation θ = θ 0 Circle centered at the origin: A circle of radius r 0 centered at the origin has equation r = r 0 in polar coordinates. Example Graph the following equations r = 5, θ = The equation r = 5 describes a circle of radius 5 centered at the origin. The equation θ = describes a line through the origin making an angle with the positive x axis. of
19 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ 0 r 3 5
20 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ 0 3 r We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). 5
21 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ 0 3 r We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). 5
22 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ r 0 0 6/ 6 3 6/ 0 5 6/ 3 6 We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 0 θ, since (r(θ), θ) = (r(θ + ), θ + ).
23 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ r 0 0 6/ 6 3 6/ 0 5 6/ 3 6 We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). When we plot the points, we see that they lie on a circle of radius 3 centered at (0, 3) and that the curve traces out this circle twice in an anticlockwise direction as θ increases from 0 to
24 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ r 0 0 6/ 6 3 6/ 0 5 6/ 3 6 We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). When we plot the points, we see that they lie on a circle of radius 3 centered at (0, 3) and that the curve traces out this circle twice in an anticlockwise direction as θ increases from 0 to The equation in Cart. coords is x + (y 3) =
25 Example 5 Example 5 Graph the equation r = 6 sin θ and convert the equation to an equation in Cartesian coordinates. θ r Cartesian Coords (x, y) 0 0 (0, 0) 6/ (3, 3) 6 (0, 6) 3 6/ ( 3, 3) 0 (0, 0) 5 6/ (3, 3) 3 6 (0, 6) It may help to calculate the cartesian coordinates in order to sketch the curve.
26 Example 6 Example 6 Graph the equation r = 1 + cos θ. Check the variations shown at end of lecture notes. θ r Cartesian Coords (x, y)
27 Example 6 Example 6 Graph the equation r = 1 + cos θ. Check the variations shown at end of lecture notes. θ r Cartesian Coords (x, y) 0 (, 0) +1 ( +1, +1 ) 1 (0, 1) 3 1 ( ( 1), 1 ) 0 (0, 0) 5 1 ( ( 1), ( 1) ) 3 1 (0, 1) 7 +1 ( ( +1), ( +1) ) (, 0) 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval
28 Example 6 Example 6 Graph the equation r = 1 + cos θ. Check the variations shown at end of lecture notes. θ r Cartesian Coords (x, y) 0 (, 0) +1 ( +1, +1 ) 1 (0, 1) 3 1 ( ( 1), 1 ) 0 (0, 0) 5 1 ( ( 1), ( 1) ) 3 1 (0, 1) 7 +1 ( ( +1), ( +1) ) (, 0) 0 θ, since (r(θ), θ) = (r(θ + ), θ + ). When we plot the points, we see that they lie on a heart shaped curve We find the value of r for specific values of θ in order to plot some points on the curve. We note that it is enough to sketch the graph on the interval 5 3 7
29 Using Symmetry We have 3 common symmetries in curves which often shorten our work in graphing a curve of the form r = f (θ) : If f ( θ) = f (θ) the curve is symmetric about the horizontal line θ = 0. In this case it is enough to draw either the upper or lower half of the curve, drawing the other half by reflecting in the line θ = 0(the xaxis). [Example r = 1 + cos θ]
30 Using Symmetry If f (θ) = f (θ + ), the curve has central symmetry about the pole or origin. Here it is enough to draw the graph in either the upper or right half plane and then rotate by 180 degree to get the other half. [Example r = sin θ.]
31 Using Symmetry If f (θ) = f ( θ), the curve is symmetric about the vertical line θ =. It is enough to draw either the right half or the left half of the curve in this case. [Example r = 6 sin θ.]
32 Example (Symmetry) Example 7 Sketch the rose r = cos(θ)
33 Example (Symmetry) Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis.
34 Example (Symmetry) Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis. Since cos(θ) = cos(( θ)) = cos( θ), we have symmetry with respect to the xaxis.
35 Example (Symmetry) Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis. Since cos(θ) = cos(( θ)) = cos( θ), we have symmetry with respect to the xaxis. Therefore it is enough to draw part of the graph for 0 θ and find the rest of the graph (for 0 θ or θ ) by symmetry.
36 Example (Symmetry) Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis. Since cos(θ) = cos(( θ)) = cos( θ), we have symmetry with respect to the xaxis. Therefore it is enough to draw part of the graph for 0 θ and find the rest of the graph (for 0 θ or θ ) by symmetry. θ 0 1 r = cos(θ)
37 Example (Symmetry) Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis. Since cos(θ) = cos(( θ)) = cos( θ), we have symmetry with respect to the xaxis. Therefore it is enough to draw part of the graph for 0 θ and find the rest of the graph (for 0 θ or θ ) by symmetry. θ 0 1 r = cos(θ)
38 Example7 Example 7 Sketch the rose r = cos(θ) Since cos(θ) = cos(( θ)), we have symmetry with respect to the xaxis. Since cos(θ) = cos(( θ)) = cos( θ), we have symmetry with respect to the xaxis. Therefore it is enough to draw part of the graph for 0 θ and find the rest of the graph (for 0 θ or θ ) by symmetry We reflect in the yaxis to get the graph for 0 θ and subsequently in the x axis to get the graph for θ.
39 Circles We have many equations of circles with polar coordinates: r = a is the circle centered at the origin of radius a, r = a sin θ is the circle of radius a centered at (a, ) (on the yaxis), and r = a cos θ is the circle of radius a centered at (a, 0) (on the xaxis). Below, we show the graphs of r =, r = sin θ and r = cos θ
40 Tangents to Polar Curves If we want to find the equation of a tangent line to a curve of the form r = f (θ), we write the equation of the curve in parametric form, using the parameter θ. x = r cos θ = f (θ) cos θ, y = r sin θ = f (θ) sin θ From the calculus of parametric equations, we know that if f is differentiable and continuous we have the formula: dy dy dx = dθ = f dr (θ) sin θ + f (θ) cos θ f (θ) cos θ f (θ) sin θ = sin θ + r cos θ dθ cos θ r sin θ dx dθ Note As usual, we locate horizontal tangents by identifying the points where dy/dx = 0 and we locate vertical tangents by identifying the points where dy/dx = dr dθ
41 Tangents to Polar Curves If we want to find the equation of a tangent line to a curve of the form r = f (θ), we write the equation of the curve in parametric form, using the parameter θ. x = r cos θ = f (θ) cos θ, y = r sin θ = f (θ) sin θ From the calculus of parametric equations, we know that if f is differentiable and continuous we have the formula: dy dy dx = dθ = f dr (θ) sin θ + f (θ) cos θ f (θ) cos θ f (θ) sin θ = sin θ + r cos θ dθ cos θ r sin θ dx dθ Note As usual, we locate horizontal tangents by identifying the points where dy/dx = 0 and we locate vertical tangents by identifying the points where dy/dx = dr dθ
42 Example: Tangents to Polar Curves Example 8 Find the equation of the tangent to the curve r = θ when θ =
43 Example: Tangents to Polar Curves Example 8 Find the equation of the tangent to the curve r = θ when θ = We have parametric equations x = θ cos θ and y = θ sin θ.
44 Example: Tangents to Polar Curves Example 8 Find the equation of the tangent to the curve r = θ when θ = We have parametric equations x = θ cos θ and y = θ sin θ. dy = dy/dθ sin θ+θ cos θ = dx dx/dθ cos θ θ sin θ
45 Example: Tangents to Polar Curves Example 8 Find the equation of the tangent to the curve r = θ when θ = We have parametric equations x = θ cos θ and y = θ sin θ. dy = dy/dθ sin θ+θ cos θ = dx dx/dθ cos θ θ sin θ When θ =, dy dx = =.
46 Example: Tangents to Polar Curves Example 8 Find the equation of the tangent to the curve r = θ when θ = We have parametric equations x = θ cos θ and y = θ sin θ. dy = dy/dθ sin θ+θ cos θ = dx dx/dθ cos θ θ sin θ When θ =, dy dx = =. When θ =, the corresponding point on the curve in polar coordinates is given by (, ) and in Cartesian coordinates by (0, ).
47 Example 8 Find the equation of the tangent to the curve r = θ when θ = We have parametric equations x = θ cos θ and y = θ sin θ. and dy = dy/dθ sin θ+θ cos θ = dx dx/dθ cos θ θ sin θ When θ =, dy dx = =. When θ =, the corresponding point on the curve in polar coordinates is given by (, ) and in Cartesian coordinates by (0, ). Therefore the equation of the tangent line, when θ = is given by (y ) = x.
48 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 (b) At which points do we have a horizontal tangent? (c) At which points do we have a vertical tangent?
49 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. (b) At which points do we have a horizontal tangent? (c) At which points do we have a vertical tangent?
50 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = 1 sin θ cos θ sin θ cos θ = 6 cos θ 6 sin θ cos θ sin θ (b) At which points do we have a horizontal tangent? (c) At which points do we have a vertical tangent?
51 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. (b) At which points do we have a horizontal tangent? (c) At which points do we have a vertical tangent?
52 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. When θ =, x = 3 3 and y = 9. Therefore the equation of the tangent 3 at this point is y 9 = 3(x 3 3 ). (b) At which points do we have a horizontal tangent? (c) At which points do we have a vertical tangent?
53 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. When θ =, x = 3 3 and y = 9. Therefore the equation of the tangent 3 at this point is y 9 = 3(x 3 3 ). (b) At which points do we have a horizontal tangent? The tangent is horizontal when dy/dθ = 0 (as long as dx/dθ 0). (c) At which points do we have a vertical tangent?
54 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. When θ =, x = 3 3 and y = 9. Therefore the equation of the tangent 3 at this point is y 9 = 3(x 3 3 ). (b) At which points do we have a horizontal tangent? The tangent is horizontal when dy/dθ = 0 (as long as dx/dθ 0). dy/dθ = 0 if 1 sin θ cos θ = 0, this happens if either sin θ = 0 or cos θ = 0. That is if θ = n/ for any integer n. (Note dx/dt 0 at any of these poinrs). (c) At which points do we have a vertical tangent?
55 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. When θ =, x = 3 3 and y = 9. Therefore the equation of the tangent 3 at this point is y 9 = 3(x 3 3 ). (b) At which points do we have a horizontal tangent? The tangent is horizontal when dy/dθ = 0 (as long as dx/dθ 0). dy/dθ = 0 if 1 sin θ cos θ = 0, this happens if either sin θ = 0 or cos θ = 0. That is if θ = n/ for any integer n. (Note dx/dt 0 at any of these poinrs). (c) At which points do we have a vertical tangent? The tangent is vertical when dx/dθ = 0 (as long as dy/dθ 0).
56 Example 9 (a) Find the equation of the tangent to the circle r = 6 sin θ when θ = 3 In parametric form, we have x = r cos θ = 6 sin θ cos θ and y = r sin θ = 6 sin θ. We have dy dx = dy/dθ dx/dθ = When θ = dy, we have 3 1 sin θ cos θ 6 cos θ 6 sin θ = 3/ dx = sin θ cos θ cos θ sin θ (1/) (3/) = 3. When θ =, x = 3 3 and y = 9. Therefore the equation of the tangent 3 at this point is y 9 = 3(x 3 3 ). (b) At which points do we have a horizontal tangent? The tangent is horizontal when dy/dθ = 0 (as long as dx/dθ 0). dy/dθ = 0 if 1 sin θ cos θ = 0, this happens if either sin θ = 0 or cos θ = 0. That is if θ = n/ for any integer n. (Note dx/dt 0 at any of these poinrs). (c) At which points do we have a vertical tangent? The tangent is vertical when dx/dθ = 0 (as long as dy/dθ 0). This happens if cos θ = sin θ, which is true if cos θ = ± sin θ. True if θ = (n + 1).
57 Example
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