Precalculus Examples

$(6 \sqrt{2}, \frac{11 π}{6})$

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 $r = 6 \sqrt{2}$ and $θ = \frac{11 π}{6}$ into the formulas.

$x = (6 \sqrt{2}) \cos (\frac{11 π}{6})$

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = 6 \sqrt{2} \cos (\frac{π}{6})$

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 4

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

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

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

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 5.2

Cancel the common factor.

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 5.3

Rewrite the expression.

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 6

Combine using the product rule for radicals.

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

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 7

Multiply $3$ by $2$ .

$x = 3 \sqrt{6}$

$y = (6 \sqrt{2}) \sin (\frac{11 π}{6})$

Step 8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$x = 3 \sqrt{6}$

$y = 6 \sqrt{2} (- \sin (\frac{π}{6}))$

Step 9

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

$x = 3 \sqrt{6}$

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = 3 \sqrt{6}$

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

Step 10.2

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

$x = 3 \sqrt{6}$

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

Step 10.3

Cancel the common factor.

$x = 3 \sqrt{6}$

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

Step 10.4

Rewrite the expression.

$x = 3 \sqrt{6}$

$y = 3 \sqrt{2} \cdot - 1$

$x = 3 \sqrt{6}$

$y = 3 \sqrt{2} \cdot - 1$

Step 11

Multiply $- 1$ by $3$ .

$x = 3 \sqrt{6}$

$y = - 3 \sqrt{2}$

Step 12

The rectangular representation of the polar point $(6 \sqrt{2}, \frac{11 π}{6})$ is $(3 \sqrt{6}, - 3 \sqrt{2})$ .

$(3 \sqrt{6}, - 3 \sqrt{2})$