Precalculus Examples

$(4, \frac{- 3 π}{4})$

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

$x = (4) \cos (\frac{- 3 π}{4})$

$y = (4) \sin (\frac{- 3 π}{4})$

Step 3

Move the negative in front of the fraction.

$x = 4 \cos (- \frac{3 π}{4})$

$y = (4) \sin (\frac{- 3 π}{4})$

Step 4

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

$x = 4 \cos (\frac{5 π}{4})$

$y = (4) \sin (\frac{- 3 π}{4})$

Step 5

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.

$x = 4 (- \cos (\frac{π}{4}))$

$y = (4) \sin (\frac{- 3 π}{4})$

Step 6

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

$y = (4) \sin (\frac{- 3 π}{4})$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

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

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

$y = (4) \sin (\frac{- 3 π}{4})$

Step 7.2

Factor $2$ out of $4$ .

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

$y = (4) \sin (\frac{- 3 π}{4})$

Step 7.3

Cancel the common factor.

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

$y = (4) \sin (\frac{- 3 π}{4})$

Step 7.4

Rewrite the expression.

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

$y = (4) \sin (\frac{- 3 π}{4})$

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

$y = (4) \sin (\frac{- 3 π}{4})$

Step 8

Multiply $- 1$ by $2$ .

$x = - 2 \sqrt{2}$

$y = (4) \sin (\frac{- 3 π}{4})$

Step 9

Move the negative in front of the fraction.

$x = - 2 \sqrt{2}$

$y = 4 \sin (- \frac{3 π}{4})$

Step 10

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

$x = - 2 \sqrt{2}$

$y = 4 \sin (\frac{5 π}{4})$

Step 11

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.

$x = - 2 \sqrt{2}$

$y = 4 (- \sin (\frac{π}{4}))$

Step 12

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$x = - 2 \sqrt{2}$

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

Step 13

Cancel the common factor of $2$ .

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Step 13.1

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

$x = - 2 \sqrt{2}$

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

Step 13.2

Factor $2$ out of $4$ .

$x = - 2 \sqrt{2}$

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

Step 13.3

Cancel the common factor.

$x = - 2 \sqrt{2}$

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

Step 13.4

Rewrite the expression.

$x = - 2 \sqrt{2}$

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

$x = - 2 \sqrt{2}$

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

Step 14

Multiply $- 1$ by $2$ .

$x = - 2 \sqrt{2}$

$y = - 2 \sqrt{2}$

Step 15

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

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