Trigonometry Examples

$(6, \frac{π}{6})$

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 $r = 6$ and $θ = \frac{π}{6}$ into the formulas.

$x = (6) \cos (\frac{π}{6})$

$y = (6) \sin (\frac{π}{6})$

Step 3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$x = 6 (\frac{\sqrt{3}}{2})$

$y = (6) \sin (\frac{π}{6})$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $6$ .

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

$y = (6) \sin (\frac{π}{6})$

Step 4.2

Cancel the common factor.

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

$y = (6) \sin (\frac{π}{6})$

Step 4.3

Rewrite the expression.

$x = 3 \sqrt{3}$

$y = (6) \sin (\frac{π}{6})$

$x = 3 \sqrt{3}$

$y = (6) \sin (\frac{π}{6})$

Step 5

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$x = 3 \sqrt{3}$

$y = 6 (\frac{1}{2})$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $6$ .

$x = 3 \sqrt{3}$

$y = 2 (3) (\frac{1}{2})$

Step 6.2

Cancel the common factor.

$x = 3 \sqrt{3}$

$y = 2 \cdot (3 (\frac{1}{2}))$

Step 6.3

Rewrite the expression.

$x = 3 \sqrt{3}$

$y = 3$

$x = 3 \sqrt{3}$

$y = 3$

Step 7

The rectangular representation of the polar point $(6, \frac{π}{6})$ is $(3 \sqrt{3}, 3)$ .

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