Precalculus Examples

$(2, 210 °)$

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 $r = 2$ and $θ = 210 °$ into the formulas.

$x = (2) \cos (210 °)$

$y = (2) \sin (210 °)$

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.

$x = 2 (- \cos (30))$

$y = (2) \sin (210 °)$

Step 4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

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

$y = (2) \sin (210 °)$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

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

$y = (2) \sin (210 °)$

Step 5.2

Cancel the common factor.

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

$y = (2) \sin (210 °)$

Step 5.3

Rewrite the expression.

$x = - \sqrt{3}$

$y = (2) \sin (210 °)$

$x = - \sqrt{3}$

$y = (2) \sin (210 °)$

Step 6

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.

$x = - \sqrt{3}$

$y = 2 (- \sin (30))$

Step 7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$x = - \sqrt{3}$

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = - \sqrt{3}$

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

Step 8.2

Cancel the common factor.

$x = - \sqrt{3}$

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

Step 8.3

Rewrite the expression.

$x = - \sqrt{3}$

$y = - 1$

$x = - \sqrt{3}$

$y = - 1$

Step 9

The rectangular representation of the polar point $(2, 210 °)$ is $(- \sqrt{3}, - 1)$ .

$(- \sqrt{3}, - 1)$