Convert to Rectangular Coordinates (8,(7pi)/6)

$(8, \frac{7 π}{6})$

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 $r = 8$ and $θ = \frac{7 π}{6}$ into the formulas.

$x = (8) \cos (\frac{7 π}{6})$

$y = (8) \sin (\frac{7 π}{6})$

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 third quadrant.

$x = 8 (- \cos (\frac{π}{6}))$

$y = (8) \sin (\frac{7 π}{6})$

Step 4

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

$x = 8 (- \frac{\sqrt{3}}{2})$

$y = (8) \sin (\frac{7 π}{6})$

Step 5

Cancel the common factor of $2$ .

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

$x = 8 (\frac{- \sqrt{3}}{2})$

$y = (8) \sin (\frac{7 π}{6})$

Factor $2$ out of $8$ .

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

$y = (8) \sin (\frac{7 π}{6})$

Cancel the common factor.

$x = 2 \cdot (4 (\frac{- \sqrt{3}}{2}))$

$y = (8) \sin (\frac{7 π}{6})$

Rewrite the expression.

$x = 4 (- \sqrt{3})$

$y = (8) \sin (\frac{7 π}{6})$

$x = 4 (- \sqrt{3})$

$y = (8) \sin (\frac{7 π}{6})$

Step 6

Multiply $- 1$ by $4$ .

$x = - 4 \sqrt{3}$

$y = (8) \sin (\frac{7 π}{6})$

Step 7

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 third quadrant.

$x = - 4 \sqrt{3}$

$y = 8 (- \sin (\frac{π}{6}))$

Step 8

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

$x = - 4 \sqrt{3}$

$y = 8 (- \frac{1}{2})$

Step 9

Cancel the common factor of $2$ .

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

$x = - 4 \sqrt{3}$

$y = 8 (\frac{- 1}{2})$

Factor $2$ out of $8$ .

$x = - 4 \sqrt{3}$

$y = 2 (4) (\frac{- 1}{2})$

Cancel the common factor.

$x = - 4 \sqrt{3}$

$y = 2 \cdot (4 (\frac{- 1}{2}))$

Rewrite the expression.

$x = - 4 \sqrt{3}$

$y = 4 \cdot - 1$

$x = - 4 \sqrt{3}$

$y = 4 \cdot - 1$

Step 10

Multiply $4$ by $- 1$ .

$x = - 4 \sqrt{3}$

$y = - 4$

Step 11

The rectangular representation of the polar point $(8, \frac{7 π}{6})$ is $(- 4 \sqrt{3}, - 4)$ .

$(- 4 \sqrt{3}, - 4)$

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