Trigonometry Examples

$(4 \sqrt{2}, \frac{5 π}{4})$

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

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

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

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.

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

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

Step 4

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

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

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

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

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

Step 5.2

Factor $2$ out of $4 \sqrt{2}$ .

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

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

Step 5.3

Cancel the common factor.

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

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

Step 5.4

Rewrite the expression.

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

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

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

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

Step 6

Multiply $- 1$ by $2$ .

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

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

Step 7

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

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

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

Step 8

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

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

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

Step 9

Add $1$ and $1$ .

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

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

Step 10

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

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Step 10.1

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

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

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

Step 10.2

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

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

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

Step 10.3

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

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

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

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

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

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

Step 10.4.2

Rewrite the expression.

$x = - 2 \cdot 2$

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

$x = - 2 \cdot 2$

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

Step 10.5

Evaluate the exponent.

$x = - 2 \cdot 2$

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

$x = - 2 \cdot 2$

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

Step 11

Multiply $- 2$ by $2$ .

$x = - 4$

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

Step 12

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.

$x = - 4$

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

Step 13

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

$x = - 4$

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

Step 14

Cancel the common factor of $2$ .

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Step 14.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$x = - 4$

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

Step 14.2

Factor $2$ out of $4 \sqrt{2}$ .

$x = - 4$

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

Step 14.3

Cancel the common factor.

$x = - 4$

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

Step 14.4

Rewrite the expression.

$x = - 4$

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

$x = - 4$

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

Step 15

Multiply $- 1$ by $2$ .

$x = - 4$

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

Step 16

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

$x = - 4$

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

Step 17

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

$x = - 4$

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

Step 18

Add $1$ and $1$ .

$x = - 4$

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

Step 19

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

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Step 19.1

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

$x = - 4$

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

Step 19.2

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

$x = - 4$

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

Step 19.3

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

$x = - 4$

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

Step 19.4

Cancel the common factor of $2$ .

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Step 19.4.1

Cancel the common factor.

$x = - 4$

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

Step 19.4.2

Rewrite the expression.

$x = - 4$

$y = - 2 \cdot 2$

$x = - 4$

$y = - 2 \cdot 2$

Step 19.5

Evaluate the exponent.

$x = - 4$

$y = - 2 \cdot 2$

$x = - 4$

$y = - 2 \cdot 2$

Step 20

Multiply $- 2$ by $2$ .

$x = - 4$

$y = - 4$

Step 21

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

$(- 4, - 4)$