Algebra Examples

$(\sqrt{2}, \frac{π}{4})$

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 $r = \sqrt{2}$ and $θ = \frac{π}{4}$ into the formulas.

$x = (\sqrt{2}) \cos (\frac{π}{4})$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Step 3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Step 4

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

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

$x = \frac{\sqrt{2} \sqrt{2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{\sqrt{2} \sqrt{2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{\sqrt{2} \sqrt{2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{{\sqrt{2}}^{1 + 1}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Add $1$ and $1$ .

$x = \frac{{\sqrt{2}}^{2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

$x = \frac{{\sqrt{2}}^{2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Step 5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{2^{\frac{1}{2} \cdot 2}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{2^{\frac{2}{2}}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$x = \frac{2^{\frac{2}{2}}}{2}$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Rewrite the expression.

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Evaluate the exponent.

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

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

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Step 6

Divide $2$ by $2$ .

$x = 1$

$y = (\sqrt{2}) \sin (\frac{π}{4})$

Step 7

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$x = 1$

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

Step 8

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

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

$x = 1$

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

Raise $\sqrt{2}$ to the power of $1$ .

$x = 1$

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

Raise $\sqrt{2}$ to the power of $1$ .

$x = 1$

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 1$

$y = \frac{{\sqrt{2}}^{1 + 1}}{2}$

Add $1$ and $1$ .

$x = 1$

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

$x = 1$

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

Step 9

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = 1$

$y = \frac{{(2^{\frac{1}{2}})}^{2}}{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 1$

$y = \frac{2^{\frac{1}{2} \cdot 2}}{2}$

Combine $\frac{1}{2}$ and $2$ .

$x = 1$

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

Cancel the common factor of $2$ .

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Cancel the common factor.

$x = 1$

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

Rewrite the expression.

$x = 1$

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

$x = 1$

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

Evaluate the exponent.

$x = 1$

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

$x = 1$

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

Step 10

Divide $2$ by $2$ .

$x = 1$

$y = 1$

Step 11

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

$(1, 1)$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$