Convert to Rectangular Coordinates (6,(5pi)/6)

$(6, \frac{5 π}{6})$

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 $r = 6$ and $θ = \frac{5 π}{6}$ into the formulas.

$x = (6) \cos (\frac{5 π}{6})$

$y = (6) \sin (\frac{5 π}{6})$

Step 3

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

$x = 6 (- \cos (\frac{π}{6}))$

$y = (6) \sin (\frac{5 π}{6})$

Step 4

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

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

$y = (6) \sin (\frac{5 π}{6})$

Step 5

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

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

$y = (6) \sin (\frac{5 π}{6})$

Factor $2$ out of $6$ .

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

$y = (6) \sin (\frac{5 π}{6})$

Cancel the common factor.

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

$y = (6) \sin (\frac{5 π}{6})$

Rewrite the expression.

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

$y = (6) \sin (\frac{5 π}{6})$

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

$y = (6) \sin (\frac{5 π}{6})$

Step 6

Multiply $- 1$ by $3$ .

$x = - 3 \sqrt{3}$

$y = (6) \sin (\frac{5 π}{6})$

Step 7

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

$x = - 3 \sqrt{3}$

$y = 6 \sin (\frac{π}{6})$

Step 8

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

$x = - 3 \sqrt{3}$

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

Step 9

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$x = - 3 \sqrt{3}$

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

Cancel the common factor.

$x = - 3 \sqrt{3}$

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

Rewrite the expression.

$x = - 3 \sqrt{3}$

$y = 3$

$x = - 3 \sqrt{3}$

$y = 3$

Step 10

The rectangular representation of the polar point $(6, \frac{5 π}{6})$ is $(- 3 \sqrt{3}, 3)$ .

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

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