Calculus Examples

$(\frac{2}{3}, - \frac{2 π}{3})$

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

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

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

Step 3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = \frac{2}{3} \cdot \cos (\frac{4 π}{3})$

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

Step 4

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.

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

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

Step 5

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

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

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

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

$x = \frac{2}{3} \cdot \frac{- 1}{2}$

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

Step 6.2

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{- 1}{2}$

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

Step 6.3

Rewrite the expression.

$x = \frac{1}{3} \cdot - 1$

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

$x = \frac{1}{3} \cdot - 1$

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

Step 7

Combine $\frac{1}{3}$ and $- 1$ .

$x = \frac{- 1}{3}$

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

Step 8

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

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

Step 9

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = - \frac{1}{3}$

$y = \frac{2}{3} \cdot \sin (\frac{4 π}{3})$

Step 10

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.

$x = - \frac{1}{3}$

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

Step 11

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

$x = - \frac{1}{3}$

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

Step 12

Cancel the common factor of $2$ .

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Step 12.1

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

$x = - \frac{1}{3}$

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

Step 12.2

Cancel the common factor.

$x = - \frac{1}{3}$

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

Step 12.3

Rewrite the expression.

$x = - \frac{1}{3}$

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

$x = - \frac{1}{3}$

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

Step 13

Combine $\frac{1}{3}$ and $\sqrt{3}$ .

$x = - \frac{1}{3}$

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

Step 14

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

$(- \frac{1}{3}, - \frac{\sqrt{3}}{3})$