Precalculus Examples

$(- 2, \frac{4 π}{3})$

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

$x = (- 2) \cos (\frac{4 π}{3})$

$y = (- 2) \sin (\frac{4 π}{3})$

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 third quadrant.

$x = - 2 (- \cos (\frac{π}{3}))$

$y = (- 2) \sin (\frac{4 π}{3})$

Step 4

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

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

$y = (- 2) \sin (\frac{4 π}{3})$

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.

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

$y = (- 2) \sin (\frac{4 π}{3})$

Step 5.2

Factor $2$ out of $- 2$ .

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

$y = (- 2) \sin (\frac{4 π}{3})$

Step 5.3

Cancel the common factor.

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

$y = (- 2) \sin (\frac{4 π}{3})$

Step 5.4

Rewrite the expression.

$x = - 1 \cdot - 1$

$y = (- 2) \sin (\frac{4 π}{3})$

$x = - 1 \cdot - 1$

$y = (- 2) \sin (\frac{4 π}{3})$

Step 6

Multiply $- 1$ by $- 1$ .

$x = 1$

$y = (- 2) \sin (\frac{4 π}{3})$

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.

$x = 1$

$y = - 2 (- \sin (\frac{π}{3}))$

Step 8

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

$x = 1$

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

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

$x = 1$

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

Step 9.2

Factor $2$ out of $- 2$ .

$x = 1$

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

Step 9.3

Cancel the common factor.

$x = 1$

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

Step 9.4

Rewrite the expression.

$x = 1$

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

$x = 1$

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

Step 10

Multiply.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$x = 1$

$y = 1 \sqrt{3}$

Step 10.2

Multiply $\sqrt{3}$ by $1$ .

$x = 1$

$y = \sqrt{3}$

$x = 1$

$y = \sqrt{3}$

Step 11

The rectangular representation of the polar point $(- 2, \frac{4 π}{3})$ is $(1, \sqrt{3})$ .

$(1, \sqrt{3})$