Question

transform each polar equation to an equation in rectanglular coordinates and identify its shape. a. θ=1.34 radians b. r=tanθsecθ

Answer 1

x=tan(1.34) sec(1.34) cos(1.34)
y=tan(1.34) sec(1.34) sin(1.34)

Answer 2

in converting polar coordinates to rectangular coordinates, use the following:
\[x = r*cos(\theta)\]
\[y = r*\sin(\theta)\]
\[\theta = \tan^{-1}(\frac{y}{x})\]
Now given is
\[r = \tan(\theta)*\sec(\theta)\]
we also know that
\[r = \frac{x}{\cos(\theta)} = \frac{y}{\sin(\theta)}\]
By substitution
\[\frac{x}{\cos(\theta)} = \tan(\theta)*\sec(\theta)\]
\[x = \tan(\theta)\]
By substitution
\[\frac{y}{\sin(\theta)} = \frac{x}{\cos(\theta)}\]
\[y = x*\tan(\theta) =x^{2}\]
Therefore the rectangular form of this equation is y=x^2 which is a parabola.
when \[\theta = 1.34\]
\[r = \tan(1.34)*\sec(1.34) = 18.6\]
\[x = \tan(1.34) = 4.256\]
\[y = x^{2} = 4.256^{2} = 18.11\]
The polar coordinates of the point are (1.34,18.06)
The rectangular coordinates are (4.256,18.11)