Question

How do you convert \[\theta =\dfrac{5\pi }{6}\] to rectangular form?

Answer 1

Hint: In this problem, we have to find the rectangular form of the given polar form. We know the relation between the polar form and the cartesian form. By using the relation between the polar form and the cartesian form we can divide the both x and y value to get a tangent. We can substitute the given angle in the tangent to get the rectangular form.

Complete step-by-step answer:
We know that the given angle is,
\[\theta =\dfrac{5\pi }{6}\]…….. (1)
We also know that the relation between the polar form and the cartesian form is,
\[\begin{align}
  & x=r\cos \theta ......(2) \\
 & y=r\sin \theta .......(3) \\
\end{align}\]
We can now divide (3) and (2) we get
\[\Rightarrow \dfrac{\sin \theta }{\cos \theta }=\dfrac{y}{x}\]
We can write the above step as,
\[\Rightarrow \tan \theta =\dfrac{y}{x}\]
We can now substitute the given angle (1) in the above step, we get
\[\Rightarrow \tan \dfrac{5\pi }{6}=\dfrac{y}{x}\]
We know that \[\tan \dfrac{5\pi }{6}=-\dfrac{1}{\sqrt{3}}\], we can substitute this in the above step, we get
\[\Rightarrow \dfrac{y}{x}=-\dfrac{1}{\sqrt{3}}\]
We can now multiply x on both left-hand side and the right-hand side of the equation, we get
\[\Rightarrow y=-\dfrac{1}{\sqrt{3}}x\]
Therefore, the rectangular form of the polar form \[\theta =\dfrac{5\pi }{6}\] is \[y=-\dfrac{1}{\sqrt{3}}x\].
We can now plot the graph for the rectangular form.

Note: Students make mistakes while writing the tangent value for the given angle. We should know some trigonometric degree values and formulas such as dividing sin and cosine to get a tangent, to be used in these types of problems. We should also remember the relation between the polar form and the rectangular form.