Precalculus Examples

$(4, 225 °)$

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 $θ = 225 °$ into the formulas.

$x = (4) \cos (225 °)$

$y = (4) \sin (225 °)$

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.

$x = 4 (- \cos (45))$

$y = (4) \sin (225 °)$

Step 4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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

$y = (4) \sin (225 °)$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

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

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

$y = (4) \sin (225 °)$

Step 5.2

Factor $2$ out of $4$ .

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

$y = (4) \sin (225 °)$

Step 5.3

Cancel the common factor.

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

$y = (4) \sin (225 °)$

Step 5.4

Rewrite the expression.

$x = 2 (- \sqrt{2})$

$y = (4) \sin (225 °)$

$x = 2 (- \sqrt{2})$

$y = (4) \sin (225 °)$

Step 6

Multiply $- 1$ by $2$ .

$x = - 2 \sqrt{2}$

$y = (4) \sin (225 °)$

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.

$x = - 2 \sqrt{2}$

$y = 4 (- \sin (45))$

Step 8

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 2 \sqrt{2}$

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

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

$x = - 2 \sqrt{2}$

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

Step 9.2

Factor $2$ out of $4$ .

$x = - 2 \sqrt{2}$

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

Step 9.3

Cancel the common factor.

$x = - 2 \sqrt{2}$

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

Step 9.4

Rewrite the expression.

$x = - 2 \sqrt{2}$

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

$x = - 2 \sqrt{2}$

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

Step 10

Multiply $- 1$ by $2$ .

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

Step 11

The rectangular representation of the polar point $(4, 225 °)$ is $(- 2 \sqrt{2}, - 2 \sqrt{2})$ .

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