Question

How do you convert \[r=\sec \theta \] into cartesian form?

Answer 1

Hint: From the given question we are given to convert \[r=\sec \theta \] into cartesian form. For that we have to assume the given equation as equation (1) and we have to simplify the equation (1) using trigonometric conversions. After simplifying the equation apply the conversion formula to get the cartesian form.

Complete step by step answer:
From the given question, we are given to convert \[r=\sec \theta \] into cartesian form.
So let us convert the equation into cartesian form by considering it as equation (1).
Let us consider
\[r=\sec \theta ............\left( 1 \right)\]
Let us divide with \[\sec \theta \]on both sides, we get
\[\dfrac{r}{\sec \theta }=1\]
Let us consider the above equation as equation (2).
\[\dfrac{r}{\sec \theta }=1...........\left( 2 \right)\]
As we all know\[\dfrac{1}{\sec \theta }=\cos \theta \].
Let us consider the above formula as formula ($\f_1$).
\[\dfrac{1}{\sec \theta }=\cos \theta .........\left( $\f_1$ \right)\]
Now, let us apply formula ($f_1$) to equation (2), we get
\[r\left( \cos \theta \right)=1\]
\[\Rightarrow r\cos \theta =1\]
So, let us consider the above equation as equation (3).
\[r\cos \theta =1............\left( 3 \right)\]
Now, by the conversion formula;
\[x=r\cos \theta ,y=r\sin \theta \]
Let us consider the above formula as ($f_2$) and ($f_3$).
\[\begin{align}
  & x=r\cos \theta ...............\left( $\f_2$ \right) \\
 & y=r\sin \theta ................\left( $\f_3$ \right) \\
\end{align}\]
By observing equation (3) and formula ($f_2$), we can apply formula ($f_2$) to equation (3).
Applying formula ($f_2$) to equation (3), we get
\[x=1\]
Let us consider the above equation as equation (4)
\[x=1...........\left( 4 \right)\]

So, therefore by converting the equation \[r=\sec \theta \] to cartesian form we will get \[x=1\].

Note: For solving this problem students should have deep knowledge on concept polar coordinates. Students should have keen knowledge on trigonometric notations too. Examiner may ask to convert a complex equation to cartesian form like\[r=\dfrac{{{\cos }^{2}}x.{{\sec }^{2}}x}{\cos ecx}\] to solve this type of equations we should know the trigonometric formulas perfectly.