Precalculus Examples

$(- 4.25, - \frac{5 π}{6})$

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 $r = - 4.25$ and $θ = - \frac{5 π}{6}$ into the formulas.

$x = (- 4.25) \cos (- \frac{5 π}{6})$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 3

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

$x = - 4.25 \cos (\frac{7 π}{6})$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 4

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.

$x = - 4.25 (- \cos (\frac{π}{6}))$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 5

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

$x = - 4.25 (- \frac{\sqrt{3}}{2})$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 6

Multiply $- 4.25 (- \frac{\sqrt{3}}{2})$ .

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Step 6.1

Multiply $- 1$ by $- 4.25$ .

$x = 4.25 (\frac{\sqrt{3}}{2})$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 6.2

Combine $4.25$ and $\frac{\sqrt{3}}{2}$ .

$x = \frac{4.25 \sqrt{3}}{2}$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 6.3

Multiply $4.25$ by $\sqrt{3}$ .

$x = \frac{7.36121593}{2}$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

$x = \frac{7.36121593}{2}$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 7

Divide $7.36121593$ by $2$ .

$x = 3.68060796$

$y = (- 4.25) \sin (- \frac{5 π}{6})$

Step 8

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

$x = 3.68060796$

$y = - 4.25 \sin (\frac{7 π}{6})$

Step 9

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.

$x = 3.68060796$

$y = - 4.25 (- \sin (\frac{π}{6}))$

Step 10

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

$x = 3.68060796$

$y = - 4.25 (- \frac{1}{2})$

Step 11

Multiply $- 4.25 (- \frac{1}{2})$ .

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Step 11.1

Multiply $- 1$ by $- 4.25$ .

$x = 3.68060796$

$y = 4.25 (\frac{1}{2})$

Step 11.2

Combine $4.25$ and $\frac{1}{2}$ .

$x = 3.68060796$

$y = \frac{4.25}{2}$

$x = 3.68060796$

$y = \frac{4.25}{2}$

Step 12

Divide $4.25$ by $2$ .

$x = 3.68060796$

$y = 2.125$

Step 13

The rectangular representation of the polar point $(- 4.25, - \frac{5 π}{6})$ is $(3.68060796, 2.125)$ .

$(3.68060796, 2.125)$