Q 34 - Calculus

Question

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems 

Change $\theta =\alpha$, where $\alpha$ is a constant, to the rectangular system

Step-by-Step Solution

Answer

The given equation in rectangular system is $x\sin \alpha -y\cos \alpha =0$

Step 1:Given information

The given expression is $\theta =\alpha$

Step 2:Simplification

Converting the equation $\theta =\alpha$ in the recatangular system as below

Since $\tan \theta =\frac{y}{x}$

$\Rightarrow \theta ={\tan }^{-1}\frac{y}{x}$

Now,

substituting in $\theta =\alpha$

${\tan }^{-1}\frac{y}{x}=\alpha$

$\Rightarrow \frac{y}{x}=\tan \alpha$

$\Rightarrow \frac{y}{x}=\frac{\sin \alpha }{\cos \alpha }$

$\Rightarrow x\sin \alpha -y\cos \alpha =0$

Therefore, the given expression in rectangular system is $x\sin \alpha -y\cos \alpha =0$