Question

If $(x,y)$are the rectangular coordinates of a point P and $(r,\theta )$are its polar coordinate, then$x=\_\_\_\_\_\_\_$and$y=\_\_\_\_\_\_$.

Short answer

$$\begin{gathered} x=r\cos \left(\theta \right) \\ y=r\sin \left(\theta \right) \end{gathered}$$

Step-by-step solution

Step 1. Given information

Rectangular coordinate at P is $(x,y)$

Polar coordinate at P is$(r,\theta )$

Step 2. Sketch the triangle using point P

Locate sides of the right-angled triangle

opposite side is$y$

adjacent side is$x$

hypotenuse side is$r$

angle is$\theta$

Now, we can locate these values into the triangle

Step 3. Finding y-coordinate

Apply the sine formula

$\sin \left(\theta \right)=\frac{opp}{hyp}$

Plug opposite and hypotenuse value

role="math" localid="1646761071652" $$\begin{gathered} \sin \left(\theta \right)=\frac{y}{r} \\ r\times \frac{y}{r}=r\times \sin \left(\theta \right) \\ y=r\sin \left(\theta \right) \end{gathered}$$

Step 3. Finding x-coordinate

Apply the cosine formula

$\cos \left(\theta \right)=\frac{adj}{hyp}$

Plug adjacent and hypotenuse value

$$\begin{gathered} \cos \left(\theta \right)=\frac{x}{r} \\ \frac{x}{r}=\cos \left(\theta \right) \\ r\times \frac{x}{r}=r\times \cos \left(\theta \right) \\ x=r\cos \left(\theta \right) \end{gathered}$$