Convert to Rectangular Coordinates (-9,120 degrees )

$(- 9, 120 °)$

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 $r = - 9$ and $θ = 120 °$ into the formulas.

$x = (- 9) \cos (120 °)$

$y = (- 9) \sin (120 °)$

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.

$x = - 9 (- \cos (60))$

$y = (- 9) \sin (120 °)$

Step 4

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$x = - 9 (- \frac{1}{2})$

$y = (- 9) \sin (120 °)$

Step 5

Multiply $- 9 (- \frac{1}{2})$ .

Multiply $- 1$ by $- 9$ .

$x = 9 (\frac{1}{2})$

$y = (- 9) \sin (120 °)$

Combine $9$ and $\frac{1}{2}$ .

$x = \frac{9}{2}$

$y = (- 9) \sin (120 °)$

$x = \frac{9}{2}$

$y = (- 9) \sin (120 °)$

Step 6

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

$x = \frac{9}{2}$

$y = - 9 \sin (60)$

Step 7

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$x = \frac{9}{2}$

$y = - 9 \frac{\sqrt{3}}{2}$

Step 8

Combine $- 9$ and $\frac{\sqrt{3}}{2}$ .

$x = \frac{9}{2}$

$y = \frac{- 9 \sqrt{3}}{2}$

Step 9

Move the negative in front of the fraction.

$x = \frac{9}{2}$

$y = - \frac{9 \sqrt{3}}{2}$

Step 10

The rectangular representation of the polar point $(- 9, 120 °)$ is $(\frac{9}{2}, - \frac{9 \sqrt{3}}{2})$ .

$(\frac{9}{2}, - \frac{9 \sqrt{3}}{2})$

Convert to Rectangular Coordinates (-9,120 degrees )

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