Precalculus Examples

$(5 + \sqrt{5}, - 3 + \sqrt{8})$

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 $r = 5 + \sqrt{5}$ and $θ = - 3 + \sqrt{8}$ into the formulas.

$x = (5 + \sqrt{5}) \cos (- 3 + \sqrt{8})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 3

Simplify each term.

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Step 3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.1.1

Factor $4$ out of $8$ .

$x = (5 + \sqrt{5}) \cos (- 3 + \sqrt{4 (2)})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 3.1.2

Rewrite $4$ as $2^{2}$ .

$x = (5 + \sqrt{5}) \cos (- 3 + \sqrt{2^{2} \cdot 2})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

$x = (5 + \sqrt{5}) \cos (- 3 + \sqrt{2^{2} \cdot 2})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 3.2

Pull terms out from under the radical.

$x = (5 + \sqrt{5}) \cos (- 3 + 2 \sqrt{2})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

$x = (5 + \sqrt{5}) \cos (- 3 + 2 \sqrt{2})$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 4

Evaluate $\cos (- 3 + 2 \sqrt{2})$ .

$x = (5 + \sqrt{5}) \cdot 0.98531744$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 5

Multiply $5 + \sqrt{5}$ by $0.98531744$ .

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{8})$

Step 6

Simplify each term.

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Step 6.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.1

Factor $4$ out of $8$ .

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{4 (2)})$

Step 6.1.2

Rewrite $4$ as $2^{2}$ .

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{2^{2} \cdot 2})$

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + \sqrt{2^{2} \cdot 2})$

Step 6.2

Pull terms out from under the radical.

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + 2 \sqrt{2})$

$x = 7.12982401$

$y = (5 + \sqrt{5}) \sin (- 3 + 2 \sqrt{2})$

Step 7

Evaluate $\sin (- 3 + 2 \sqrt{2})$ .

$x = 7.12982401$

$y = (5 + \sqrt{5}) \cdot - 0.17073234$

Step 8

Multiply $5 + \sqrt{5}$ by $- 0.17073234$ .

$x = 7.12982401$

$y = - 1.23543082$

Step 9

The rectangular representation of the polar point $(5 + \sqrt{5}, - 3 + \sqrt{8})$ is $(7.12982401, - 1.23543082)$ .

$(7.12982401, - 1.23543082)$