Calculus Examples

$(4, \frac{7 π}{3})$

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 $r = 4$ and $θ = \frac{7 π}{3}$ into the formulas.

$x = (4) \cos (\frac{7 π}{3})$

$y = (4) \sin (\frac{7 π}{3})$

Step 3

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

$x = 4 \cos (\frac{π}{3})$

$y = (4) \sin (\frac{7 π}{3})$

Step 4

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

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

$y = (4) \sin (\frac{7 π}{3})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $4$ .

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

$y = (4) \sin (\frac{7 π}{3})$

Step 5.2

Cancel the common factor.

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

$y = (4) \sin (\frac{7 π}{3})$

Step 5.3

Rewrite the expression.

$x = 2$

$y = (4) \sin (\frac{7 π}{3})$

$x = 2$

$y = (4) \sin (\frac{7 π}{3})$

Step 6

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

$x = 2$

$y = 4 \sin (\frac{π}{3})$

Step 7

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

$x = 2$

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4$ .

$x = 2$

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

Step 8.2

Cancel the common factor.

$x = 2$

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

Step 8.3

Rewrite the expression.

$x = 2$

$y = 2 \sqrt{3}$

$x = 2$

$y = 2 \sqrt{3}$

Step 9

The rectangular representation of the polar point $(4, \frac{7 π}{3})$ is $(2, 2 \sqrt{3})$ .

$(2, 2 \sqrt{3})$