9.3 – Polar & Rectangular Forms of Equations Objectives: 1. Convert between polar and rectangular coordinates. 2. Convert between polar and rectangular equations.

Convert Polar to Rectangular Coordinates:

Example 1: A. Find the rectangular coordinates for B. Find the rectangular coordinates for F(–5, 45°).

Example 1: C. Find the rectangular coordinates for H(4, –240°). D. Find the rectangular coordinates for R(–8, 300°).

Convert Rectangular to Polar Coordinates:

Example 2: A. Find two pairs of polar coordinates for the point E(2, –4). B. Find two pairs of polar coordinates for the point G(–2, –4).

Example 2: C. Find two pairs of polar coordinates for F(–5, –6) with the given rectangular coordinates.

Example 3: HIDDEN TREASURE A crew is using radar to search for pirate treasure hidden under water. Suppose the boat is facing due east, and the radar gives the polar coordinates of the treasure as (8, 205o). What are the rectangular coordinates for the hidden treasure?

Rectangular Equations to Polar Equations: Replace x with 𝑟 cos 𝜃 and y with 𝑟 sin 𝜃 . Simplify using algebra and trig. identities.

Example 4: A. Write the rectangular equation in polar form. (x + 2) 2 + y 2 = 4

Example 4: B. Write the rectangular equation in polar form. 2xy = 4

Example 4: C. Write the rectangular equation in polar form. 𝒙 𝟐 + 𝒚 𝟐 +𝟐𝒚=𝟎

Polar Equations to Rectangular Equations: If just 𝜃, tan both sides. Get 𝑟 2 if necessary. Replace 𝑟 2 with 𝑥 2 + 𝑦 2 . Replace 𝑟 cos 𝜃 with x and 𝑟 sin 𝜃 with y.

Example 5: A. Write the polar equation in rectangular form.

Example 5: B. Write the polar equation in rectangular form. r = 5

Example 5: C. Write the polar equation in rectangular form. 𝒓 𝟐 =𝟐𝒓 𝒄𝒐𝒔 𝜽

Example 5: D. Write the polar equation in rectangular form. r = 2 sin 𝜽