Precalculus Examples

$(8, - \frac{2 π}{3})$

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 $r = 8$ and $θ = - \frac{2 π}{3}$ into the formulas.

$x = (8) \cos (- \frac{2 π}{3})$

$y = (8) \sin (- \frac{2 π}{3})$

Step 3

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

$x = 8 \cos (\frac{4 π}{3})$

$y = (8) \sin (- \frac{2 π}{3})$

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.

$x = 8 (- \cos (\frac{π}{3}))$

$y = (8) \sin (- \frac{2 π}{3})$

Step 5

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

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

$y = (8) \sin (- \frac{2 π}{3})$

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.

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

$y = (8) \sin (- \frac{2 π}{3})$

Step 6.2

Factor $2$ out of $8$ .

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

$y = (8) \sin (- \frac{2 π}{3})$

Step 6.3

Cancel the common factor.

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

$y = (8) \sin (- \frac{2 π}{3})$

Step 6.4

Rewrite the expression.

$x = 4 \cdot - 1$

$y = (8) \sin (- \frac{2 π}{3})$

$x = 4 \cdot - 1$

$y = (8) \sin (- \frac{2 π}{3})$

Step 7

Multiply $4$ by $- 1$ .

$x = - 4$

$y = (8) \sin (- \frac{2 π}{3})$

Step 8

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

$x = - 4$

$y = 8 \sin (\frac{4 π}{3})$

Step 9

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.

$x = - 4$

$y = 8 (- \sin (\frac{π}{3}))$

Step 10

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

$x = - 4$

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

Step 11

Cancel the common factor of $2$ .

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Step 11.1

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

$x = - 4$

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

Step 11.2

Factor $2$ out of $8$ .

$x = - 4$

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

Step 11.3

Cancel the common factor.

$x = - 4$

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

Step 11.4

Rewrite the expression.

$x = - 4$

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

$x = - 4$

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

Step 12

Multiply $- 1$ by $4$ .

$x = - 4$

$y = - 4 \sqrt{3}$

Step 13

The rectangular representation of the polar point $(8, - \frac{2 π}{3})$ is $(- 4, - 4 \sqrt{3})$ .

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