Precalculus Examples

$(40, 1500)$

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 $r = 40$ and $θ = 1500$ into the formulas.

$x = (40) \cos (1500)$

$y = (40) \sin (1500)$

Step 3

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$x = 40 \cos (60)$

$y = (40) \sin (1500)$

Step 4

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$x = 40 (\frac{1}{2})$

$y = (40) \sin (1500)$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $40$ .

$x = 2 (20) (\frac{1}{2})$

$y = (40) \sin (1500)$

Step 5.2

Cancel the common factor.

$x = 2 \cdot (20 (\frac{1}{2}))$

$y = (40) \sin (1500)$

Step 5.3

Rewrite the expression.

$x = 20$

$y = (40) \sin (1500)$

$x = 20$

$y = (40) \sin (1500)$

Step 6

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$x = 20$

$y = 40 \sin (60)$

Step 7

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$x = 20$

$y = 40 (\frac{\sqrt{3}}{2})$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $40$ .

$x = 20$

$y = 2 (20) (\frac{\sqrt{3}}{2})$

Step 8.2

Cancel the common factor.

$x = 20$

$y = 2 \cdot (20 (\frac{\sqrt{3}}{2}))$

Step 8.3

Rewrite the expression.

$x = 20$

$y = 20 \sqrt{3}$

$x = 20$

$y = 20 \sqrt{3}$

Step 9

The rectangular representation of the polar point $(40, 1500)$ is $(20, 20 \sqrt{3})$ .

$(20, 20 \sqrt{3})$