Trigonometry Examples

$(- 2, - \frac{4 π}{3})$

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 $θ = - \frac{4 π}{3}$ into the formulas.

$x = (- 2) \cos (- \frac{4 π}{3})$

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 3

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = - 2 \cos (\frac{2 π}{3})$

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 4

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.

$x = - 2 (- \cos (\frac{π}{3}))$

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 5

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 6.2

Factor $2$ out of $- 2$ .

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

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 6.3

Cancel the common factor.

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

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 6.4

Rewrite the expression.

$x = - 1 \cdot - 1$

$y = (- 2) \sin (- \frac{4 π}{3})$

$x = - 1 \cdot - 1$

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 7

Multiply $- 1$ by $- 1$ .

$x = 1$

$y = (- 2) \sin (- \frac{4 π}{3})$

Step 8

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$x = 1$

$y = - 2 \sin (\frac{2 π}{3})$

Step 9

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

$x = 1$

$y = - 2 \sin (\frac{π}{3})$

Step 10

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$x = 1$

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

Step 11

Cancel the common factor of $2$ .

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Step 11.1

Factor $2$ out of $- 2$ .

$x = 1$

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

Step 11.2

Cancel the common factor.

$x = 1$

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

Step 11.3

Rewrite the expression.

$x = 1$

$y = - 1 \sqrt{3}$

$x = 1$

$y = - 1 \sqrt{3}$

Step 12

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$x = 1$

$y = - \sqrt{3}$

Step 13

The rectangular representation of the polar point $(- 2, - \frac{4 π}{3})$ is $(1, - \sqrt{3})$ .

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