Download Trigonometry: Polar Coordinates

Trigonometry:
Polar Coordinates
Honors Precalculus
Mr. Velazquez
Plotting Points with Polar Coordinates
Plotting Points with Polar Coordinates
y
Plot the points with
the following polar
coordinates:
(a) (4, 30°)
(b)
3𝜋
3,
4
(c) (−2, 120°)
x





Representations of Points in Polar Coords
Representations of Points in Polar Coords
𝜋
The point 3, 6 is plotted in the figure. Find another representation of the
point in which:
(a) r is positive, and 2𝜋 ≤ 𝜃 < 4𝜋
y
(b) r is negative, and 0 ≤ 𝜃 < 2𝜋
x

(c) r is negative, and −2𝜋 ≤ 𝜃 < 0




Polar and Rectangular Coordinates
Polar and Rectangular Coordinates
Polar and Rectangular Coordinates
Find the rectangular
coordinates of the
points with the
following polar
coordinates:
(a)
y
𝜋
3,
2
x

(b)
2𝜋
−2,
3




Polar and Rectangular Coordinates
Convert the rectangular point (−1, 3) to polar
coordinates:
Polar and Rectangular Coordinates
Find the polar
coordinates of the
points with the
following rectangular
coordinates:
(a)
y
3 , −1
x

(b) −2, 2 3




Equation Conversion: Rectangular to Polar
We can turn an equation in rectangular form to an equation in
polar form in the following steps:
1. Substitute 𝒙 = 𝒓 𝐜𝐨𝐬 𝜽 and 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 for 𝑥 and 𝑦.
2. Solve the resulting equation for 𝑟.
Convert the rectangular equation to a polar equation
that expresses r in terms of  .
y  3x  2
Equation Conversion: Rectangular to Polar
We can turn an equation in rectangular form to an equation in
polar form in the following steps:
1. Substitute 𝒙 = 𝒓 𝐜𝐨𝐬 𝜽 and 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 for 𝑥 and 𝑦.
2. Solve the resulting equation for 𝑟.
Convert the rectangular equation to a polar equation
that expresses r in terms of  .
( x  3) 2 +(y+2) 2  1
Equation Conversion: Polar to Rectangular
We can turn an equation in polar form to an equation in
rectangular form with the following substiutions:
𝑟2 = 𝑥2 + 𝑦2
𝑟 cos 𝜃 = 𝑥
𝑟 sin 𝜃 = 𝑦
Convert the polar equation to a rectangular
equation in x and y. r=3
tan 𝜃 =
𝑦
𝑥
Equation Conversion: Polar to Rectangular
We can turn an equation in polar form to an equation in
rectangular form with the following substiutions:
𝑟2 = 𝑥2 + 𝑦2
𝑟 cos 𝜃 = 𝑥
𝑟 sin 𝜃 = 𝑦
Convert the polar equation to a rectangular
equation in x and y.  =

6
tan 𝜃 =
𝑦
𝑥
Equation Conversion: Polar to Rectangular
We can turn an equation in polar form to an equation in
rectangular form with the following substiutions:
𝑟2 = 𝑥2 + 𝑦2
𝑟 cos 𝜃 = 𝑥
𝑟 sin 𝜃 = 𝑦
Convert the polar equation to a rectangular
equation in x and y. r= -2 cosx
tan 𝜃 =
𝑦
𝑥
Equation Conversion: Polar to Rectangular
We can turn an equation in polar form to an equation in
rectangular form with the following substiutions:
𝑟2 = 𝑥2 + 𝑦2
𝑟 cos 𝜃 = 𝑥
𝑟 sin 𝜃 = 𝑦
Convert the polar equation to a rectangular
equation in x and y. r= 5 csc 
tan 𝜃 =
𝑦
𝑥
Exit Ticket: Polar Coordinates
1. Convert the following equations from rectangular to polar form:
2. Convert the following equations from polar to rectangular form:
HOMEWORK (DUE 3/10)
Pg. 672-673, #4-72
(multiples of 4)