Precalculus Examples

$(- 4, \frac{2 π}{3})$

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 $r = - 4$ and $θ = \frac{2 π}{3}$ into the formulas.

$x = (- 4) \cos (\frac{2 π}{3})$

$y = (- 4) \sin (\frac{2 π}{3})$

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.

$x = - 4 (- \cos (\frac{π}{3}))$

$y = (- 4) \sin (\frac{2 π}{3})$

Step 4

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

$y = (- 4) \sin (\frac{2 π}{3})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

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

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

$y = (- 4) \sin (\frac{2 π}{3})$

Step 5.2

Factor $2$ out of $- 4$ .

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

$y = (- 4) \sin (\frac{2 π}{3})$

Step 5.3

Cancel the common factor.

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

$y = (- 4) \sin (\frac{2 π}{3})$

Step 5.4

Rewrite the expression.

$x = - 2 \cdot - 1$

$y = (- 4) \sin (\frac{2 π}{3})$

$x = - 2 \cdot - 1$

$y = (- 4) \sin (\frac{2 π}{3})$

Step 6

Multiply $- 2$ by $- 1$ .

$x = 2$

$y = (- 4) \sin (\frac{2 π}{3})$

Step 7

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

$x = 2$

$y = - 4 \sin (\frac{π}{3})$

Step 8

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

$x = 2$

$y = - 4 \frac{\sqrt{3}}{2}$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $- 4$ .

$x = 2$

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

Step 9.2

Cancel the common factor.

$x = 2$

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

Step 9.3

Rewrite the expression.

$x = 2$

$y = - 2 \sqrt{3}$

$x = 2$

$y = - 2 \sqrt{3}$

Step 10

The rectangular representation of the polar point $(- 4, \frac{2 π}{3})$ is $(2, - 2 \sqrt{3})$ .

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