Precalculus Examples

$(3, \frac{- 5 π}{6})$

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 $r = 3$ and $θ = \frac{- 5 π}{6}$ into the formulas.

$x = (3) \cos (\frac{- 5 π}{6})$

$y = (3) \sin (\frac{- 5 π}{6})$

Step 3

Move the negative in front of the fraction.

$x = 3 \cos (- \frac{5 π}{6})$

$y = (3) \sin (\frac{- 5 π}{6})$

Step 4

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

$x = 3 \cos (\frac{7 π}{6})$

$y = (3) \sin (\frac{- 5 π}{6})$

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.

$x = 3 (- \cos (\frac{π}{6}))$

$y = (3) \sin (\frac{- 5 π}{6})$

Step 6

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

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

$y = (3) \sin (\frac{- 5 π}{6})$

Step 7

Multiply $3 (- \frac{\sqrt{3}}{2})$ .

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Step 7.1

Multiply $- 1$ by $3$ .

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

$y = (3) \sin (\frac{- 5 π}{6})$

Step 7.2

Combine $- 3$ and $\frac{\sqrt{3}}{2}$ .

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

$y = (3) \sin (\frac{- 5 π}{6})$

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

$y = (3) \sin (\frac{- 5 π}{6})$

Step 8

Move the negative in front of the fraction.

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

$y = (3) \sin (\frac{- 5 π}{6})$

Step 9

Move the negative in front of the fraction.

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

$y = 3 \sin (- \frac{5 π}{6})$

Step 10

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

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

$y = 3 \sin (\frac{7 π}{6})$

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.

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

$y = 3 (- \sin (\frac{π}{6}))$

Step 12

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

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

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

Step 13

Multiply $3 (- \frac{1}{2})$ .

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Step 13.1

Multiply $- 1$ by $3$ .

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

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

Step 13.2

Combine $- 3$ and $\frac{1}{2}$ .

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

$y = \frac{- 3}{2}$

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

$y = \frac{- 3}{2}$

Step 14

Move the negative in front of the fraction.

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

$y = - \frac{3}{2}$

Step 15

The rectangular representation of the polar point $(3, \frac{- 5 π}{6})$ is $(- \frac{3 \sqrt{3}}{2}, - \frac{3}{2})$ .

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