Trigonometry Examples

$(3 \sqrt{2}, \frac{5 π}{4})$

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 $r = 3 \sqrt{2}$ and $θ = \frac{5 π}{4}$ into the formulas.

$x = (3 \sqrt{2}) \cos (\frac{5 π}{4})$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

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 third quadrant.

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 4

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

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5

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

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Step 5.1

Multiply $- 1$ by $3$ .

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.2

Combine $\frac{\sqrt{2}}{2}$ and $- 3$ .

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.3

Combine $\frac{\sqrt{2} \cdot - 3}{2}$ and $\sqrt{2}$ .

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.4

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

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.5

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

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.6

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

$x = \frac{- 3 {\sqrt{2}}^{1 + 1}}{2}$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 5.7

Add $1$ and $1$ .

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6

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

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Step 6.1

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

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6.2

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

$x = \frac{- 3 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6.3

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

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6.4.2

Rewrite the expression.

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 6.5

Evaluate the exponent.

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

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

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $- 3$ by $2$ .

$x = \frac{- 6}{2}$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 7.2

Divide $- 6$ by $2$ .

$x = - 3$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

$x = - 3$

$y = (3 \sqrt{2}) \sin (\frac{5 π}{4})$

Step 8

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.

$x = - 3$

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

Step 9

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

$x = - 3$

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

Step 10

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

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Step 10.1

Multiply $- 1$ by $3$ .

$x = - 3$

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

Step 10.2

Combine $\frac{\sqrt{2}}{2}$ and $- 3$ .

$x = - 3$

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

Step 10.3

Combine $\frac{\sqrt{2} \cdot - 3}{2}$ and $\sqrt{2}$ .

$x = - 3$

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

Step 10.4

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

$x = - 3$

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

Step 10.5

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

$x = - 3$

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

Step 10.6

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

$x = - 3$

$y = \frac{- 3 {\sqrt{2}}^{1 + 1}}{2}$

Step 10.7

Add $1$ and $1$ .

$x = - 3$

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

$x = - 3$

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

Step 11

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

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Step 11.1

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

$x = - 3$

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

Step 11.2

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

$x = - 3$

$y = \frac{- 3 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

Step 11.3

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

$x = - 3$

$y = \frac{- 3 \cdot 2^{\frac{2}{2}}}{2}$

Step 11.4

Cancel the common factor of $2$ .

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Step 11.4.1

Cancel the common factor.

$x = - 3$

$y = \frac{- 3 \cdot 2^{\frac{2}{2}}}{2}$

Step 11.4.2

Rewrite the expression.

$x = - 3$

$y = \frac{- 3 \cdot 2}{2}$

$x = - 3$

$y = \frac{- 3 \cdot 2}{2}$

Step 11.5

Evaluate the exponent.

$x = - 3$

$y = \frac{- 3 \cdot 2}{2}$

$x = - 3$

$y = \frac{- 3 \cdot 2}{2}$

Step 12

Simplify the expression.

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Step 12.1

Multiply $- 3$ by $2$ .

$x = - 3$

$y = \frac{- 6}{2}$

Step 12.2

Divide $- 6$ by $2$ .

$x = - 3$

$y = - 3$

$x = - 3$

$y = - 3$

Step 13

The rectangular representation of the polar point $(3 \sqrt{2}, \frac{5 π}{4})$ is $(- 3, - 3)$ .

$(- 3, - 3)$