Trigonometry Examples

(- 3, \frac{π}{4})

$(- 3, \frac{π}{4})$

Step 1

Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

x = r c o s θ

$x = r c o s θ$

y = r s i n θ

$y = r s i n θ$

Step 2

Substitute in the known values of

r = - 3

$r = - 3$ and

θ = \frac{π}{4}

$θ = \frac{π}{4}$ into the formulas.

x = (- 3) cos (\frac{π}{4})

$x = (- 3) \cos (\frac{π}{4})$

y = (- 3) sin (\frac{π}{4})

$y = (- 3) \sin (\frac{π}{4})$

Step 3

The exact value of

cos (\frac{π}{4})

$\cos (\frac{π}{4})$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

y = (- 3) sin (\frac{π}{4})

$y = (- 3) \sin (\frac{π}{4})$

Step 4

Combine

- 3

$- 3$ and

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

y = (- 3) sin (\frac{π}{4})

$y = (- 3) \sin (\frac{π}{4})$

Step 5

Move the negative in front of the fraction.

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

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

y = (- 3) sin (\frac{π}{4})

$y = (- 3) \sin (\frac{π}{4})$

Step 6

The exact value of

sin (\frac{π}{4})

$\sin (\frac{π}{4})$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

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

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

Step 7

Combine

- 3

$- 3$ and

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

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

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

y = \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}

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

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

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

Step 9

The rectangular representation of the polar point

(- 3, \frac{π}{4})

$(- 3, \frac{π}{4})$ is

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

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

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

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