Trigonometry Examples

(2, \frac{π}{3})

$(2, \frac{π}{3})$

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 = 2

$r = 2$ and

θ = \frac{π}{3}

$θ = \frac{π}{3}$ into the formulas.

x = (2) \cos (\frac{π}{3})

$x = (2) \cos (\frac{π}{3})$

y = (2) \sin (\frac{π}{3})

$y = (2) \sin (\frac{π}{3})$

Step 3

The exact value of

\cos (\frac{π}{3})

$\cos (\frac{π}{3})$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

y = (2) \sin (\frac{π}{3})

$y = (2) \sin (\frac{π}{3})$

Step 4

Cancel the common factor of

2

$2$ .

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Step 4.1

Cancel the common factor.

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

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

y = (2) \sin (\frac{π}{3})

$y = (2) \sin (\frac{π}{3})$

Step 4.2

Rewrite the expression.

x = 1

$x = 1$

y = (2) \sin (\frac{π}{3})

$y = (2) \sin (\frac{π}{3})$

x = 1

$x = 1$

y = (2) \sin (\frac{π}{3})

$y = (2) \sin (\frac{π}{3})$

Step 5

The exact value of

\sin (\frac{π}{3})

$\sin (\frac{π}{3})$ is

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

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

x = 1

$x = 1$

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

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

Step 6

Cancel the common factor of

2

$2$ .

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Step 6.1

Cancel the common factor.

x = 1

$x = 1$

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

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

Step 6.2

Rewrite the expression.

x = 1

$x = 1$

y = \sqrt{3}

$y = \sqrt{3}$

x = 1

$x = 1$

y = \sqrt{3}

$y = \sqrt{3}$

Step 7

The rectangular representation of the polar point

(2, \frac{π}{3})

$(2, \frac{π}{3})$ is

(1, \sqrt{3})

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

(1, \sqrt{3})

$(1, \sqrt{3})$