Precalculus Examples

$(\frac{4 \sqrt{3}}{2}, \frac{1}{2})$

Step 1

Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

$x = r c o s θ$

$y = r s i n θ$

Step 2

Substitute in the known values of $r = \frac{4 \sqrt{3}}{2}$ and $θ = \frac{1}{2}$ into the formulas.

$x = (\frac{4 \sqrt{3}}{2}) \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 3

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $4 \sqrt{3}$ .

$x = \frac{2 (2 \sqrt{3})}{2} \cdot \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{3})}{2 (1)} \cdot \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 3.2.2

Cancel the common factor.

$x = \frac{2 (2 \sqrt{3})}{2 \cdot 1} \cdot \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 3.2.3

Rewrite the expression.

$x = \frac{2 \sqrt{3}}{1} \cdot \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 3.2.4

Divide $2 \sqrt{3}$ by $1$ .

$x = 2 \sqrt{3} \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

$x = 2 \sqrt{3} \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

$x = 2 \sqrt{3} \cos (\frac{1}{2})$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 4

Evaluate $\cos (\frac{1}{2})$ .

$x = 2 \sqrt{3} \cdot 0.87758256$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 5

Multiply $2 \sqrt{3} \cdot 0.87758256$ .

Tap for more steps...

Step 5.1

Multiply $0.87758256$ by $2$ .

$x = 1.75516512 \sqrt{3}$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 5.2

Multiply $1.75516512$ by $\sqrt{3}$ .

$x = 3.04003517$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

$x = 3.04003517$

$y = (\frac{4 \sqrt{3}}{2}) \sin (\frac{1}{2})$

Step 6

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 6.1

Factor $2$ out of $4 \sqrt{3}$ .

$x = 3.04003517$

$y = \frac{2 (2 \sqrt{3})}{2} \cdot \sin (\frac{1}{2})$

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

Factor $2$ out of $2$ .

$x = 3.04003517$

$y = \frac{2 (2 \sqrt{3})}{2 (1)} \cdot \sin (\frac{1}{2})$

Step 6.2.2

Cancel the common factor.

$x = 3.04003517$

$y = \frac{2 (2 \sqrt{3})}{2 \cdot 1} \cdot \sin (\frac{1}{2})$

Step 6.2.3

Rewrite the expression.

$x = 3.04003517$

$y = \frac{2 \sqrt{3}}{1} \cdot \sin (\frac{1}{2})$

Step 6.2.4

Divide $2 \sqrt{3}$ by $1$ .

$x = 3.04003517$

$y = 2 \sqrt{3} \sin (\frac{1}{2})$

$x = 3.04003517$

$y = 2 \sqrt{3} \sin (\frac{1}{2})$

$x = 3.04003517$

$y = 2 \sqrt{3} \sin (\frac{1}{2})$

Step 7

Evaluate $\sin (\frac{1}{2})$ .

$x = 3.04003517$

$y = 2 \sqrt{3} \cdot 0.47942553$

Step 8

Multiply $2 \sqrt{3} \cdot 0.47942553$ .

Tap for more steps...

Step 8.1

Multiply $0.47942553$ by $2$ .

$x = 3.04003517$

$y = 0.95885107 \sqrt{3}$

Step 8.2

Multiply $0.95885107$ by $\sqrt{3}$ .

$x = 3.04003517$

$y = 1.66077878$

$x = 3.04003517$

$y = 1.66077878$

Step 9

The rectangular representation of the polar point $(\frac{4 \sqrt{3}}{2}, \frac{1}{2})$ is $(3.04003517, 1.66077878)$ .

$(3.04003517, 1.66077878)$