Polar coordinates - Parametric Equations and Polar Coordinates

Polar coordinates

In this section, we will introduce a new coordinate system called polar coordinates. We will introduce some formulas and how they are derived. Then we will use these formulas to convert Cartesian equations to polar coordinates, and vice versa. We will then learn how to graph polar equations by using 2 methods. The first method is to change the polar equations to Cartesian coordinates, and the second method is to graph the polar equation using a table of values.

Related concepts:
Polar form of complex numbers
Operations on complex numbers in polar form

Lessons

Notes:

Our goal of this section is to introduce a new coordinate system called Polar Coordinates. Most of these questions will involve converting polar coordinates to Cartesian coordinates, converting Cartesian coordinates to polar coordinates, and drawing polar equations.

In Cartesian coordinates we say that the coordinate of the point is at $(x, y)$ . However in Polar Coordinates we say that the coordinate of the point is at $(r, \theta)$ .

When converting from Polar to Cartesian, we can use the following formulas:
$x=r \cos \theta$
$y=r \sin \theta$
$r^2=x^2+y^2$
$r=\sqrt{x^2+y^2}$
$\theta = \tan^{-1}(\frac{y}{x})$

Intro Lesson

Polar Coordinates Overview:
1.
Converting Cartesian equations into polar coordinates
Convert the following Cartesian equations into polar coordinates:
2.
Converting Polar equations into Cartesian coordinates
Convert the following Polar equations into Cartesian coordinates
3.
Graphing Polar Equations by changing to Cartesian coordinates
Convert the polar equations into Cartesian coordinates and then graph the equation:
4.
Graphing Polar Equations with table of values
Graph the Polar Equations using table of values:

Polar coordinates

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Intro

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Lessons

Notes:

Intro Lesson

Polar Coordinates Overview:

a)

Understanding the Cartesian to Polar Conversion Formulas

b)

Converting Cartesian equations into polar coordinates

c)

Converting polar equations into Cartesian coordinates

d)

Graphing Polar Equations

1.

Converting Cartesian equations into polar coordinates Convert the following Cartesian equations into polar coordinates:

a)

x2+y2=5x^2+y^2=5 x2+y2=5

b)

x2+4x+y2+4y+8=(tan⁡−1yx)2x^2+4x+y^2+4y+8=(\tan^{-1} \frac{y}{x})^2 x2+4x+y2+4y+8=(tan−1xy​)2

c)

x2−y3=6 x^2-y^3=6 x2−y3=6

2.

Converting Polar equations into Cartesian coordinates Convert the following Polar equations into Cartesian coordinates

a)

3r=cos⁡θ \frac{3}{r}=\cos \theta r3​=cosθ

b)

r4=1sec⁡θ+1csc⁡θ r^4=\frac{1}{\sec \theta} + \frac{1}{\csc \theta} r4=secθ1​+cscθ1​

c)

r12=tan⁡[θ(sin⁡2θ+cos⁡2θ)32] r^{\frac{1}{2}}=\tan[\theta (\sin^2 \theta + \cos^2 \theta)^{\frac{3}{2}}]r21​=tan[θ(sin2θ+cos2θ)23​]

3.

Graphing Polar Equations by changing to Cartesian coordinates Convert the polar equations into Cartesian coordinates and then graph the equation:

a)

r=5 r=5 r=5

b)

r=−3cos⁡θ r=-3 \cos \theta r=−3cosθ

c)

4sin⁡θ=2r 4 \sin \theta = \frac{2}{r} 4sinθ=r2​

4.

Graphing Polar Equations with table of values Graph the Polar Equations using table of values:

a)

r=1+cos⁡θ r=1+ \cos \thetar=1+cosθ

b)

r=2sin⁡θ r=2 \sin \theta r=2sinθ

Polar coordinates

or

Converting Cartesian equations into polar coordinates
Convert the following Cartesian equations into polar coordinates:

$x^2+y^2=5$

$x^2+4x+y^2+4y+8=(\tan^{-1} \frac{y}{x})^2$

$x^2-y^3=6$

Converting Polar equations into Cartesian coordinates
Convert the following Polar equations into Cartesian coordinates

$\frac{3}{r}=\cos \theta$

$r^4=\frac{1}{\sec \theta} + \frac{1}{\csc \theta}$

$r^{\frac{1}{2}}=\tan[\theta (\sin^2 \theta + \cos^2 \theta)^{\frac{3}{2}}]$

Graphing Polar Equations by changing to Cartesian coordinates
Convert the polar equations into Cartesian coordinates and then graph the equation:

$r=5$

$r=-3 \cos \theta$

$4 \sin \theta = \frac{2}{r}$

Graphing Polar Equations with table of values
Graph the Polar Equations using table of values:

$r=1+ \cos \theta$

$r=2 \sin \theta$