Find a Cartesian equation for the curve and identify it. r^{2}\cos(2\theta)=1

Kelly Nelson

Kelly Nelson

Answered question

2021-12-16

Find a Cartesian equation for the curve and identify it.
r2cos(2θ)=1

Answer & Explanation

Anzante2m

Anzante2m

Beginner2021-12-17Added 34 answers

Step 1 
We have: 
The polar formula
r2cos(2θ)=1 
Step 2 
Consider, 
r2cos(2θ)=1 
r2(cos2θsin2θ)=1 (cos(2θ)=cos2θsin2θ) 
The parametric equation is an example.
x=rcosθ 
y=rsinθ 
The implies, 
xr=cosθ 
yr=sinθ 
Then, 
(xr)2=cos2θ 
(yr)2=sin2θ 
Step 3 
Substitute (xr)2=cos2θ,(yr)2=sin2θr2(cos2θsin2θ)=1 
r2((xr)2(yr)2)=1 
r2(x2r2y2r2)=1 
r2(1r2(x2y2))=1 
x2y2=1 
Thus, the Cartesian equation of r2cos(2θ)=1isx2y2=1

eninsala06

eninsala06

Beginner2021-12-18Added 37 answers

Step 1 
r2cos2θ=1 
Recall that: cos2x=cos2xsin2x 
r2(cos2θsin2θ)=1 
Choose multiplication's distributive property over addition and subtraction.
r2cos2θr2sin2θ=1 
Rephrase as follows:
(rcosθ)2(rsinθ)2=1 
To convert the above equation into the Cartesian form, replace r sinθ with y and r cosθ with x 
x2y2=1 
This is a horizontal parabola centered at the origin with lobes opening to the right/left 
Answer 
x2y2=1

nick1337

nick1337

Expert2021-12-28Added 777 answers

Step 1
r2cos(2θ)=1
Given
Step 2
cos(2θ)=cos2(θ)sin2(θ
Trig identity cos(2θ)
Step 3
r2(cos2(θ)sin2(θ))=1
Rewritten equation, using substitution from previous line
Step 4
r2cos2(θ)r2sin2(θ))=1
Expansion from previous line
Step 5
x2y2=1
x=rcos(θ)
y=rsin(θ)
Step 6
Graph of x2y2=1, which is a hyperbola.
x2y2=1

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