Trigonometry Examples

$(- 1, - 45 °)$

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 $r = - 1$ and $θ = - 45 °$ into the formulas.

$x = (- 1) \cos (- 45 °)$

$y = (- 1) \sin (- 45 °)$

Step 3

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

$x = - 1 \cos (45)$

$y = (- 1) \sin (- 45 °)$

Step 4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$x = - 1 \frac{\sqrt{2}}{2}$

$y = (- 1) \sin (- 45 °)$

Step 5

Rewrite $- 1 \frac{\sqrt{2}}{2}$ as $- \frac{\sqrt{2}}{2}$ .

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

$y = (- 1) \sin (- 45 °)$

Step 6

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 fourth quadrant.

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

$y = - 1 (- \sin (45))$

Step 7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

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

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

Step 8

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

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Step 8.1

Multiply $- 1$ by $- 1$ .

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

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

Step 8.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

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

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

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

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

Step 9

The rectangular representation of the polar point $(- 1, - 45 °)$ is $(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ .

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