Trigonometry Examples

$(- 2, 135 °)$

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 $θ = 135 °$ into the formulas.

$x = (- 2) \cos (135 °)$

$y = (- 2) \sin (135 °)$

Step 3

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.

$x = - 2 (- \cos (45))$

$y = (- 2) \sin (135 °)$

Step 4

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

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

$y = (- 2) \sin (135 °)$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

$y = (- 2) \sin (135 °)$

Step 5.2

Factor $2$ out of $- 2$ .

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

$y = (- 2) \sin (135 °)$

Step 5.3

Cancel the common factor.

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

$y = (- 2) \sin (135 °)$

Step 5.4

Rewrite the expression.

$x = - 1 (- \sqrt{2})$

$y = (- 2) \sin (135 °)$

$x = - 1 (- \sqrt{2})$

$y = (- 2) \sin (135 °)$

Step 6

Multiply.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$x = 1 \sqrt{2}$

$y = (- 2) \sin (135 °)$

Step 6.2

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

$x = \sqrt{2}$

$y = (- 2) \sin (135 °)$

$x = \sqrt{2}$

$y = (- 2) \sin (135 °)$

Step 7

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

$x = \sqrt{2}$

$y = - 2 \sin (45)$

Step 8

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

$x = \sqrt{2}$

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $- 2$ .

$x = \sqrt{2}$

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

Step 9.2

Cancel the common factor.

$x = \sqrt{2}$

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

Step 9.3

Rewrite the expression.

$x = \sqrt{2}$

$y = - 1 \sqrt{2}$

$x = \sqrt{2}$

$y = - 1 \sqrt{2}$

Step 10

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

$x = \sqrt{2}$

$y = - \sqrt{2}$

Step 11

The rectangular representation of the polar point $(- 2, 135 °)$ is $(\sqrt{2}, - \sqrt{2})$ .

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