Precalculus Examples

$(- 1, \frac{- 3 p}{4})$

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 $r = - 1$ and $θ = \frac{- 3 p}{4}$ into the formulas.

$x = (- 1) \cos (\frac{- 3 p}{4})$

$y = (- 1) \sin (\frac{- 3 p}{4})$

Step 3

Move the negative in front of the fraction.

$x = - 1 \cos (- \frac{3 p}{4})$

$y = (- 1) \sin (\frac{- 3 p}{4})$

Step 4

Since $\cos (- \frac{3 p}{4})$ is an even function, rewrite $\cos (- \frac{3 p}{4})$ as $\cos (\frac{3 p}{4})$ .

$x = - 1 \cos (\frac{3 p}{4})$

$y = (- 1) \sin (\frac{- 3 p}{4})$

Step 5

Rewrite $- 1 \cos (\frac{3 p}{4})$ as $- \cos (\frac{3 p}{4})$ .

$x = - \cos (\frac{3 p}{4})$

$y = (- 1) \sin (\frac{- 3 p}{4})$

Step 6

Move the negative in front of the fraction.

$x = - \cos (\frac{3 p}{4})$

$y = - 1 \sin (- \frac{3 p}{4})$

Step 7

Since $\sin (- \frac{3 p}{4})$ is an odd function, rewrite $\sin (- \frac{3 p}{4})$ as $- \sin (\frac{3 p}{4})$ .

$x = - \cos (\frac{3 p}{4})$

$y = - 1 (- \sin (\frac{3 p}{4}))$

Step 8

Multiply $- 1 (- \sin (\frac{3 p}{4}))$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$x = - \cos (\frac{3 p}{4})$

$y = 1 \sin (\frac{3 p}{4})$

Step 8.2

Multiply $\sin (\frac{3 p}{4})$ by $1$ .

$x = - \cos (\frac{3 p}{4})$

$y = \sin (\frac{3 p}{4})$

$x = - \cos (\frac{3 p}{4})$

$y = \sin (\frac{3 p}{4})$

Step 9

The rectangular representation of the polar point $(- 1, \frac{- 3 p}{4})$ is $(- \cos (\frac{3 p}{4}), \sin (\frac{3 p}{4}))$ .

$(- \cos (\frac{3 p}{4}), \sin (\frac{3 p}{4}))$