Unit 4: Parametric and Polar Coordinates

Official Finals Review for PreCalc Students

By: Jake Brewington and Jimmy Davey

Understanding Parametric Equations

Purpose of Using Parametric Equations

Parametric equations are used to find the graph of the inverse of a function by expressing x and y in terms of an independent, third variable (usually t).
This third variable is called the parameter, which has a Greek origin: para means "alongside" (parallel) and meter means "measure."
When converting back to x-y form, just solve for t in the first part of the parametric equation, and then plug in that value for t in the second part of the parametric equation (because it will relate to x). Therefore, the result will be in x-y form.
When finding the inverse of parametric equations, just swap the expressions which are equal to x and y.

Advantages of Using Parametric Equations

When trying to find the graph of the inverse of a complicated function, instead of swapping x and y and then solving for y in an equation, converting the equation to parametric form is much easier.
The easiest way of doing this is setting x equal to t, and therefore replacing x with t in the original equation.
In real life applications t usually represents time.
While you can graph parametric equations by hand, I recommend using a calculator because there is a PARAMETRIC mode (be aware of your window), which will prove to be more time-efficient for a Final Exam.

Parametric Functions

Eliminating the Parameter

Parametric

x = a cos(t) → x/a = cos(t) → (x/a)² = cos²t

y = b sin(t) → y/b = sin(t) → (y/b)² = sin²t

(x/a)² + (y/b)² = cos²t + sin²t

(x/a)² + (y/b)² = 1

Standard Form

The property that allowed us to convert the parametric equation to (x/a)² + (y/b)² = 1 is a Pythagorean Property: cos²t + sin²t = 1. Sometimes, it is better to see the x-y form of a function, so you can recognize it's graph easily. For example, this function is easily recognizable as an ellipse.

Your Turn: Eliminate the Parameter: x = 4 cos(t), y = 7 sin(t)

Click For Solution:

x = 4 cos(t) → x/4 = cos(t) → (x/4)² = cos²t

y = 7 sin(t) → y/7 = sin(t) → (y/7)² = sin²t

(x/4)² + (y/7)² = cos²t + sin²t

(x/4)² + (y/7)² = 1

Ellipse

Takeaway

Parametric Equations of an Ellipse

x = h + a cos t

y = k + a sin t

a and b are the x and y radii and h and k are the coordinates of the center of the ellipse

Parametrics of Conics

Reference

Here are more conic sections and some of their graphs that will be very helpful to memorize for a Final Exam:

Experiment and Learn the Properties of These Conic Sections Yourself (Link)!

*You will find that isolating the trigonometry operations is very helpful when converting from parametric to x-y form

Parametric Motion

*Disclaimer* This Section Has an Extensive Usage of Concepts from the Vectors Unit, So Please Refer to the Vectors WebPage Before Review

In the general formula x and y refer to the coordinates of the starting position of the object, v is the intial velocity or magnitude of the object, θ is the angle which the object is moving

Projectile Motion

Here is the general formula for projectile motion:

x = (vcosθ)t + x

y = (vcosθ)t + y

In many cases, you must keep in mind gravity for the y-component of the parametric equations. Gravity's effect on an object in motion is often represented through the expression of -16t².
The real life application of projectile motion is basically kinetics.

Your Turn: Projectile Motion Real Life Application: Find The Parametric Equation of the Ship's Path (From 11-5 of Parametric Equations of Moving Objects)

Click For Solution:

x = (21cos0°)t - 43

y = (13sin90°)t + 19

x = -43 +21t

y = 19 +13t

Pretty simple! Most of where the difficulty comes into play is analyzing all aspects of the situation, so remember to read directions carefully!

Cycloids

A cycloid is the path of a fixed point on the rim of a circular object in motion.
In the figure (left): to find r, you would use the formula r = v₁ + v₂ + v₃
Therefore, r = (6t)i + 6j + [6 cos(1.5π - t)]i + [6 sin(1.5π - t)]j
So the x and y components end up being:

x = 6t + 6cos(1.5π - t)

y = 6 + 6sin(1.5π - t)

Takeaway

Parametric Equations of a Cycloid

x = a(t - sin(t))

y = a(1 - cos(t))

t represents the number of radians the circular object has rolled since it's start and a represents it's radius

Connection Between Parametric and Polar Equations

Parametric

Polar

Parametric and polar equations are two different ways to represent functions which are not written in the standard x-y form.

Polar Coordinates Quick Review

Polar coordinates are a new way of plotting points or equations on a plane. Polar coordinates can be used in physics, specifically spherical physics dealing with mechanics. They can also be used when looking at a radar, or plan-position indicator. While standard algebraic equations are written in x-y form, polar equations are in terms of r (radius) and θ.

Rather than a standard coordinate plane, the polar plane is usually thought of as a circular plane. Like a standard unit circle, angles begin at the positive x-axis and rotate counter-clockwise.

Domain and range are written in terms of angle measurements. For example, in the picture above the domain is 0≤θ≤2π, because the angle begins at 0° and rotates 360° or 2π radians.

Points on the Polar Plane

Points on the polar plane are written in (r,θ) form. In the example to the left, the point is colinear to a circle with a radius of 2, so the r value is '2.' The point is rotated π/4 radians about the pole (origin of the polar plane, making the θ value 'π/4.' Keep in mind that the point (2, π/4) is the same as (-2, 5π/4), because points can be written in multiple ways on the polar plane.

Trig Values in Polar Equations

To the left is a table showing points in the equation r = sin.

So, when we graph these points, we see that it creates a circle with a radius of 1 just above the x-axis.

When we graph the same equation on Desmos, we get the same result. Since cosθ=sinθπ/2, we can conclude that the graph for r = cos(θ) would be identical to the prior, but rotated -π/2 radians.

Polar Curves

Spirals

Spirals in the polar plane can be derived from the equation r = θ.

The greater the coefficient of θ is, the greater the distance of each point will be from the pole.

Equation Above: r = 5θ

Equation Above: r = θ/2

Petaled Roses

Equations for petaled roses are in the form r=asin(nθ) or r=acos(nθ). The a value is the amplitude of the petals, or the distance from the center. If n is odd, the number of petals is n. If n is even, the number of petals is 2n.

There is a slight rotation between when the trig value is sin and when it is cos. The rotation directly varies with the n value. Using Desmos, you can see how as n increases, the rotation between roses decreases.

Cardioids

Cardioids look like circles, but curve inwards at a certain point, creating a "cusp." The equation for cardioids is r=a±asin(θ) or r=a±acos(θ)

When the equation involves sine, the cusp faces towards the y-axis, and when it includes cosine, the cusp faces towards the x-axis.

Limaçons

Limaçons are similar to cardioids, but have a loop in the center rather than a cusp. They are written in the form r=a±b⋅sin(θ) or r=a±bcos(θ). When a>b, the curve is a convex limaçon. When a<b, the curve is a standard limaçon. When a=b, the curve is a cardioid.

Example of a Convex Limaçon.

Example of a Standard Limaçon.

Note: The length of the cusp (from the intersection) is the value of |b|-|a|.

Intersections of Polar Curves

If you go back to the Polar Art project, you will remember that finding intersection points of polar curves is a bit more difficult than it seems. When given two equations, you can solve algebraically for points of intersection. Although, it is important to remember that not every point that looks like a point of intersection when the equations are graphed actually are solutions. This connects back to the different types of polar curves, as you will need to be able to identify which polar curves are involved in the given equations.

The Stranger Things World: Polar-Coordinate Plane

When converting between polar equations and x-y equations, there are 4 main conversions we can use:

x = r cos(θ)
y = r sin(θ)
x²+ y²= r²
x/y = tan(θ)

Mastering this concept can help you fully understand polar coordinates.

Your Turn: Find The Intersection

From 11-3 Intersections of Polar Curves

Click For Solution:

r = 2 + 2sinθ r = 2 - 2cosθ

2 + 2sinθ = 2 - 2cosθ

sinθ = -cosθ

Values for θ in which this is true: 3π/4, 7π/4

Plug both values back in:

r = 2 + 2sin(3π/4) r = 2 + 2sin(7π/4)

r = 2 + 2(√2/2) r = 2 - 2(√2/2)

r = 2+√2 r = 2-√2

(2+√2, 3π/4) and (2-√2, 7π/4)

Conics in the Polar Plane

Conic Equations

To the left are the two types of ways that the equations of conics can be written. The variables k, a, and b, all determine that amplitudes of conics, and also effect the type of conic under the rule of eccentricity.

When a > b, the conic is an ellipse.
When b > a, the conic is a hyperbola.
When a = b, the conic is a parabola.
When a = 0, the conic is a vertical or horizontal line.
When b = 0, the conic is a circle.
When k = 0, the conic obviously does not exist.

Complex Numbers

As you may recall from earlier in the year, or even from Algebra II, the complex numbers plane is similar to the standard coordinate plane. Complex numbers are in the form of z=a+bi. For example, the point plotted above is written in the equation z=3-2i. It will be plotted at (3,-2).

When converting complex numbers into polar form, you must find the modulas and the angle of rotation. The modulas can be found using the equation r=√ (a²+b²). The angle of rotation can be found using the equation tan(θ)=b/a.