Trigonometry Examples

$(- \frac{\sqrt{3}}{2}, - \frac{1}{2})$

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 $r = - \frac{\sqrt{3}}{2}$ and $θ = - \frac{1}{2}$ into the formulas.

$x = (- \frac{\sqrt{3}}{2}) \cos (- \frac{1}{2})$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 3

Evaluate $\cos (- \frac{1}{2})$ .

$x = - \frac{\sqrt{3}}{2} \cdot 0.87758256$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 4

Multiply $- \frac{\sqrt{3}}{2} \cdot 0.87758256$ .

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Step 4.1

Multiply $0.87758256$ by $- 1$ .

$x = - 0.87758256 \frac{\sqrt{3}}{2}$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 4.2

Combine $- 0.87758256$ and $\frac{\sqrt{3}}{2}$ .

$x = \frac{- 0.87758256 \sqrt{3}}{2}$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 4.3

Multiply $- 0.87758256$ by $\sqrt{3}$ .

$x = \frac{- 1.52001758}{2}$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

$x = \frac{- 1.52001758}{2}$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 5

Divide $- 1.52001758$ by $2$ .

$x = - 0.76000879$

$y = (- \frac{\sqrt{3}}{2}) \sin (- \frac{1}{2})$

Step 6

Evaluate $\sin (- \frac{1}{2})$ .

$x = - 0.76000879$

$y = - \frac{\sqrt{3}}{2} \cdot - 0.47942553$

Step 7

Multiply $- \frac{\sqrt{3}}{2} \cdot - 0.47942553$ .

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Step 7.1

Multiply $- 0.47942553$ by $- 1$ .

$x = - 0.76000879$

$y = 0.47942553 (\frac{\sqrt{3}}{2})$

Step 7.2

Combine $0.47942553$ and $\frac{\sqrt{3}}{2}$ .

$x = - 0.76000879$

$y = \frac{0.47942553 \sqrt{3}}{2}$

Step 7.3

Multiply $0.47942553$ by $\sqrt{3}$ .

$x = - 0.76000879$

$y = \frac{0.83038939}{2}$

$x = - 0.76000879$

$y = \frac{0.83038939}{2}$

Step 8

Divide $0.83038939$ by $2$ .

$x = - 0.76000879$

$y = 0.41519469$

Step 9

The rectangular representation of the polar point $(- \frac{\sqrt{3}}{2}, - \frac{1}{2})$ is $(- 0.76000879, 0.41519469)$ .

$(- 0.76000879, 0.41519469)$