Precalculus Examples

$(2, 3 π)$

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 $θ = 3 π$ into the formulas.

$x = (2) \cos (3 π)$

$y = (2) \sin (3 π)$

Step 3

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

$x = 2 \cos (π)$

$y = (2) \sin (3 π)$

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 second quadrant.

$x = 2 (- \cos (0))$

$y = (2) \sin (3 π)$

Step 5

The exact value of $\cos (0)$ is $1$ .

$x = 2 (- 1 \cdot 1)$

$y = (2) \sin (3 π)$

Step 6

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.1

Multiply $- 1$ by $1$ .

$x = 2 \cdot - 1$

$y = (2) \sin (3 π)$

Step 6.2

Multiply $2$ by $- 1$ .

$x = - 2$

$y = (2) \sin (3 π)$

$x = - 2$

$y = (2) \sin (3 π)$

Step 7

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

$x = - 2$

$y = 2 \sin (π)$

Step 8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = - 2$

$y = 2 \sin (0)$

Step 9

The exact value of $\sin (0)$ is $0$ .

$x = - 2$

$y = 2 \cdot 0$

Step 10

Multiply $2$ by $0$ .

$x = - 2$

$y = 0$

Step 11

The rectangular representation of the polar point $(2, 3 π)$ is $(- 2, 0)$ .

$(- 2, 0)$