Trigonometry Examples

$(- 3, \frac{π}{4})$

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 $θ = \frac{π}{4}$ into the formulas.

$x = (- 3) \cos (\frac{π}{4})$

$y = (- 3) \sin (\frac{π}{4})$

Step 3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

$y = (- 3) \sin (\frac{π}{4})$

Step 4

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

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

$y = (- 3) \sin (\frac{π}{4})$

Step 5

Move the negative in front of the fraction.

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

$y = (- 3) \sin (\frac{π}{4})$

Step 6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

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

Step 7

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

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

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

Step 8

Move the negative in front of the fraction.

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

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

Step 9

The rectangular representation of the polar point $(- 3, \frac{π}{4})$ is $(- \frac{3 \sqrt{2}}{2}, - \frac{3 \sqrt{2}}{2})$ .

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