Respuesta :
To solve this question we will make use of the basic relationship between the variables of the polar and cartesian coordinates.
They are: [tex] tan(\theta)=\frac{y}{x} [/tex]
and [tex] y=rsin(\theta) [/tex] and [tex] x=rcos(\theta) [/tex]
Let us solve the question now:
1. [tex] \theta=tan^{-1}(\frac{y}{x})= 1.34 radians [/tex]
[tex] \therefore \frac{y}{x}=tan(1.34 radians) [/tex]
[tex] \frac{y}{x}\approx4.26 [/tex]
[tex] y=4.26x [/tex]
The above is the answer for question 1. It is a straight line which passes through the origin.
2. It is given that:
[tex] r=tan(\theta)sec(\theta) [/tex]
which can be rewritten as:
[tex] r=\frac{sin(\theta)}{cos(\theta)} \times \frac{1}{cos(\theta)} [/tex]
Now, we know that: [tex] \frac{sin(\theta)}{cos(\theta)} =tan(\theta)=\frac{y}{x} [/tex]
Therefore, we get:
[tex] r=\frac{y}{x}\times \frac{1}{cos(\theta)} [/tex]
Which gives us, after cross multiplying [tex] cos(\theta) [/tex]:
[tex] rcos(\theta)=\frac{y}{x} [/tex]
[tex] x=\frac{y}{x} [/tex] (Since, [tex] rcos(\theta)=x [/tex])
Therefore, [tex] y=x^2 [/tex] is the final answer we get by cross multiplying x.
This is a parabola.
