Trigonometry Examples

(3, \frac{2 π}{3})

$(3, \frac{2 π}{3})$

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{2 π}{3}

$θ = \frac{2 π}{3}$ into the formulas.

x = (3) \cos (\frac{2 π}{3})

$x = (3) \cos (\frac{2 π}{3})$

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

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 = 3 (- \cos (\frac{π}{3}))

$x = 3 (- \cos (\frac{π}{3}))$

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

Step 4

The exact value of

\cos (\frac{π}{3})

$\cos (\frac{π}{3})$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

Step 5

Multiply

3 (- \frac{1}{2})

$3 (- \frac{1}{2})$ .

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Step 5.1

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

Step 5.2

Combine

- 3

$- 3$ and

\frac{1}{2}

$\frac{1}{2}$ .

x = \frac{- 3}{2}

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

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

x = \frac{- 3}{2}

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

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

Step 6

Move the negative in front of the fraction.

x = - \frac{3}{2}

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

y = (3) \sin (\frac{2 π}{3})

$y = (3) \sin (\frac{2 π}{3})$

Step 7

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

x = - \frac{3}{2}

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

y = 3 \sin (\frac{π}{3})

$y = 3 \sin (\frac{π}{3})$

Step 8

The exact value of

\sin (\frac{π}{3})

$\sin (\frac{π}{3})$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

x = - \frac{3}{2}

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

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

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

Step 9

Combine

3

$3$ and

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

x = - \frac{3}{2}

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

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

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

Step 10

The rectangular representation of the polar point

(3, \frac{2 π}{3})

$(3, \frac{2 π}{3})$ is

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

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

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

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