Chapter 9, Problem 67SGA

To determine

Find the rectangular form of the equation using given statement.

Answer to Problem 67SGA

The rectangular form of the equation is $y=0$ .

Explanation of Solution

Given:

The given statement is that the path of the ship represented by the equation $r\cos \left(\theta -\frac{\pi }{2}\right)+5=0$ .

Calculation:

Use difference identity of cosine.

  $\begin{array}{l}r\cos \left(\theta -\frac{\pi }{2}\right)+5=0\\ \Rightarrow r\left(\cos \theta \cos \frac{\pi }{2}+\sin \theta \sin \frac{\pi }{2}\right)=0\end{array}$

Substitute the value of $\cos \frac{\pi }{2}=0$ and $\sin \frac{\pi }{2}=1$ in the equation.

  $\begin{array}{l}\Rightarrow r\left(\cos \theta \cdot 0+\sin \theta \cdot 1\right)=0\\ \Rightarrow r\left(0+\sin \theta \right)=0\end{array}$

Use distributive property.

  $\begin{array}{l}\Rightarrow r\sin \theta =0\\ \Rightarrow y=0\end{array}$

Hence the rectangular form of the equation is $y=0$ .