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

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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