How would you find the following. A step by step process would greatly be appreciated.
1) Convert the polar equation to a rectangular equation.
r^2 = 8sin2θ
hints:
use the relations:
x=rsinθ
y=rcosθ
which means the left-hand side
=x²+y²=r²
Also
8sin2θ = 16 sinθcosθ
=16(y/r)(x/r)
=16xy/r²
Equating both sides,
x²+y² = 16xy/r²
or
(x²+y²)² = 16xy
=2xy/(x²+y²)
Thank you so much. I was stuck on the right hand side with the 8sin2θ. Not sure what to do with the 8 and 2.
This has cleared it up very well.
To convert the polar equation r^2 = 8sin^2θ to a rectangular equation, follow these steps:
Step 1: Use the trigonometric identity sin^2θ = (1 - cos2θ) / 2
- Substitute sin^2θ in the equation with (1 - cos2θ) / 2:
r^2 = 8(1 - cos2θ) / 2
Step 2: Simplify the equation
- Multiply 8 with (1 - cos2θ) / 2:
r^2 = 4 - 4cos2θ
Step 3: Use the identity cos2θ = cos^2θ - sin^2θ
- Replace cos2θ in the equation with cos^2θ - sin^2θ:
r^2 = 4 - 4(cos^2θ - sin^2θ)
Step 4: Simplify the equation
- Distribute the -4 inside the parentheses:
r^2 = 4 - 4cos^2θ + 4sin^2θ
Step 5: Use the identity cos^2θ + sin^2θ = 1
- Replace cos^2θ + sin^2θ in the equation with 1:
r^2 = 4 - 4(1) = 0
Step 6: Rewrite the equation in terms of x and y (rectangular form)
- Since x = rcosθ and y = rsinθ, substitute x^2 + y^2 for r^2 in the equation:
x^2 + y^2 = 0
The resulting rectangular equation is x^2 + y^2 = 0, which represents a single point at the origin (0, 0) in the Cartesian coordinate system.