Convert to Rectangular Coordinates (6 square root of 3,pi/6)

$(6 \sqrt{3}, \frac{π}{6})$

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

$x = r c o s θ$

$y = r s i n θ$

Substitute in the known values of $r = 6 \sqrt{3}$ and $θ = \frac{π}{6}$ into the formulas.

$x = (6 \sqrt{3}) \cos (\frac{π}{6})$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$x = 6 \sqrt{3} (\frac{\sqrt{3}}{2})$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Cancel the common factor of $2$ .

Factor $2$ out of $6 \sqrt{3}$ .

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

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Cancel the common factor.

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

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Rewrite the expression.

$x = 3 \sqrt{3} \sqrt{3}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

$x = 3 \sqrt{3} \sqrt{3}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Raise $\sqrt{3}$ to the power of $1$ .

$x = 3 (\sqrt{3} \sqrt{3})$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 3 {\sqrt{3}}^{1 + 1}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Add $1$ and $1$ .

$x = 3 {\sqrt{3}}^{2}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 3 \cdot 3^{\frac{1}{2} \cdot 2}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Combine $\frac{1}{2}$ and $2$ .

$x = 3 \cdot 3^{\frac{2}{2}}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = 3 \cdot 3^{\frac{2}{2}}$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Rewrite the expression.

$x = 3 \cdot 3$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

$x = 3 \cdot 3$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Evaluate the exponent.

$x = 3 \cdot 3$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

$x = 3 \cdot 3$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

Multiply $3$ by $3$ .

$x = 9$

$y = (6 \sqrt{3}) \sin (\frac{π}{6})$

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$x = 9$

$y = 6 \sqrt{3} (\frac{1}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $6 \sqrt{3}$ .

$x = 9$

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

Cancel the common factor.

$x = 9$

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

Rewrite the expression.

$x = 9$

$y = 3 \sqrt{3}$

$x = 9$

$y = 3 \sqrt{3}$

The rectangular representation of the polar point $(6 \sqrt{3}, \frac{π}{6})$ is $(9, 3 \sqrt{3})$ .

$(9, 3 \sqrt{3})$

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