Chapter 9-4 Polar Form of a Linear Function
OBJ: Write the polar form of a linear equation, and graph the polar form of a linear equation.
The polar form of the equation for a line 𝑙 is closely related to the
normal form, which is 𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0. We have learned
that 𝑥 = 𝑟 cos 𝜃 𝑎𝑛𝑑 𝑦 = 𝑟 sin 𝜃. The polar form of the equation
of line 𝑙 can be obtained by substituting these values into the
normal form.
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
(𝑟 cos 𝜃) cos 𝜙 + (𝑟 sin 𝜃) sin 𝜙 − 𝑝 = 0
𝑟(cos 𝜃 cos 𝜙 + sin 𝜃 sin 𝜙) − 𝑝 = 0
𝑟 cos(𝜃 − 𝜙) = 𝑝
*cos 𝜃 cos 𝜙 + sin 𝜃 sin 𝜙 = cos(𝜃 − 𝜙)
*Trig ID…remember?
In the polar form 𝜃 𝑎𝑛𝑑 𝑟 are variables, and 𝑝 𝑎𝑛𝑑 𝜙 are constants (obtained from the normal line)
We will also use some formulas from Chapter 7
RECALL…
Normal Form of a Linear Equation:
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
Standard Form to Normal Form:
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
*divide each term by ±√𝐴2 + 𝐵 2
Choose sign opposite the sign of C
Linear to Polar Steps:
1) write the linear equation in standard form (𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0)
2) Translate the standard form into normal form
i) divide each term by ±√𝐴2 + 𝐵2
ii) sign is opposite the sign of “C”
3) Find 𝜙: 𝜙 = tan−1 (𝐵⁄𝐴)
4) Find 𝑝: 𝑝 = |
𝐶
±√𝐴2 +𝐵 2
|
5) Plug the values into polar form
Ex. 1 Write in Polar form
a) 2𝑥 + 3𝑦 − 1 = 0
Ex. 2 Write in Polar form
b) 3𝑥 − 4𝑦 + 5 = 0
Polar Form to Linear Form:
1) Use the “Sum and Difference” Identity
𝑟 cos(𝜃 ± 𝜙) = 𝑟(cos 𝜃 cos 𝜙 ∓ sin 𝜃 sin 𝜙)
2) Distribute “𝑟”
3) Substitute “𝑥” for 𝑟 cos 𝜃 and “𝑦” for 𝑟 sin 𝜃
4) Put into standard form
Ex. 3 Convert the following equation to Standard Form (Rectangular)
1 = 𝑟 cos(𝜃 + 30°)
Ex. 4 Convert the following equation to Standard Form
5 = 𝑟 cos 𝜃
Ex. 5 Convert the following equation to Standard Form
𝑟 cos(𝜃 − 𝜋⁄3) − 4 = 0
HW 9-4 p.492/ 9-23 odds *(IF over 2 Days…DAY 1 is #15-23 odd
DAY 2 is #, 9,11,13