Precalculus Examples

$(\sqrt{2}, - \frac{3 π}{4})$

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

$x = (\sqrt{2}) \cos (- \frac{3 π}{4})$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 3

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

$x = \sqrt{2} \cos (\frac{5 π}{4})$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

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 third quadrant.

$x = \sqrt{2} (- \cos (\frac{π}{4}))$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 5

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 6

Multiply $\sqrt{2} (- \frac{\sqrt{2}}{2})$ .

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Step 6.1

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 6.2

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 6.3

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 6.4

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 6.5

Add $1$ and $1$ .

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7

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

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Step 7.1

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7.2

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7.3

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

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7.4.2

Rewrite the expression.

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 7.5

Evaluate the exponent.

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Cancel the common factor.

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

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 8.2

Rewrite the expression.

$x = - 1 \cdot 1$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

$x = - 1 \cdot 1$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 9

Multiply $- 1$ by $1$ .

$x = - 1$

$y = (\sqrt{2}) \sin (- \frac{3 π}{4})$

Step 10

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

$x = - 1$

$y = \sqrt{2} \sin (\frac{5 π}{4})$

Step 11

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$x = - 1$

$y = \sqrt{2} (- \sin (\frac{π}{4}))$

Step 12

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

$x = - 1$

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

Step 13

Multiply $\sqrt{2} (- \frac{\sqrt{2}}{2})$ .

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Step 13.1

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

$x = - 1$

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

Step 13.2

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

$x = - 1$

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

Step 13.3

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

$x = - 1$

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

Step 13.4

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

$x = - 1$

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

Step 13.5

Add $1$ and $1$ .

$x = - 1$

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

$x = - 1$

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

Step 14

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

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Step 14.1

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

$x = - 1$

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

Step 14.2

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

$x = - 1$

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

Step 14.3

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

$x = - 1$

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

Step 14.4

Cancel the common factor of $2$ .

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Step 14.4.1

Cancel the common factor.

$x = - 1$

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

Step 14.4.2

Rewrite the expression.

$x = - 1$

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

$x = - 1$

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

Step 14.5

Evaluate the exponent.

$x = - 1$

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

$x = - 1$

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

Step 15

Cancel the common factor of $2$ .

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Step 15.1

Cancel the common factor.

$x = - 1$

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

Step 15.2

Rewrite the expression.

$x = - 1$

$y = - 1 \cdot 1$

$x = - 1$

$y = - 1 \cdot 1$

Step 16

Multiply $- 1$ by $1$ .

$x = - 1$

$y = - 1$

Step 17

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

$(- 1, - 1)$