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Polar Coordinates: Understanding R, Theta and Representations, Lecture notes of Pre-Calculus

The basics of polar coordinates, including the concept of a polar axis, the pole, and the representation of points using r and theta. It also covers the sign of r, multiple representations of points, and the relations between polar and rectangular coordinates. Additionally, it provides examples of finding polar and rectangular coordinates.

What you will learn

  • What is the meaning of the polar axis in polar coordinates?
  • How can you find the rectangular coordinates of a point given its polar coordinates?
  • How is a point represented in polar coordinates?
  • How can you find the polar coordinates of a point given its rectangular coordinates?
  • What are the different representations of a point in polar coordinates?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Download Polar Coordinates: Understanding R, Theta and Representations and more Lecture notes Pre-Calculus in PDF only on Docsity!

Polar Coordinates

 The foundation of the polar coordinate system is a horizontal ray that extends to the right.

 The ray is called the polar axis.

 The endpoint of the ray is called the pole.

The Sign of r and a Point’s

Location in Polar Coordinates

The point 𝑃 = 𝑟, 𝜃 is located 𝑟 units from the pole. -- 𝑟 > 0, the point lies along the terminal n side of 𝜃. -- 𝑟 < 0, the point lies along the ray opposite the terminal side of 𝜃. -- 𝑟 = 0, the point lies at the pole, regardless of 𝜃.

Plot the point.

Plot the point.

Multiple Representations of

Points.

If n is any integer, the point (𝑟, 𝜃) can be represented as

𝑟, 𝜃 = (𝑟, 𝜃 + 2𝑛𝜋) or

Find 3 representations

(5,

𝜋 4 )

a. r is positive and 2𝜋 < 𝜃 < 4𝜋

b. r is negative and 0 < 𝜃 < 2𝜋

c. r is positive and −2𝜋 < 𝜃 < 0

Graph of a Circle

Relations between Polar and

Rectangular Coordinates

𝑥^2 + 𝑦^2 = 𝑟^2

r 𝑠𝑖𝑛𝜃 = y 𝑐𝑜𝑠𝜃 = θ 𝑡𝑎𝑛𝜃 = x 𝑥 = 𝑦 = 𝑡𝑎𝑛𝜃 =

Find the rectangular coordinates

What if it’s not on the Unit Circle?

Not on Unit Circle

Find the Polar Coordinates

Find the Polar Coordinates