Trigonometry Examples

$(6 \sqrt{3}, \frac{4 π}{3})$

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 $r = 6 \sqrt{3}$ and $θ = \frac{4 π}{3}$ into the formulas.

$x = (6 \sqrt{3}) \cos (\frac{4 π}{3})$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

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.

$x = 6 \sqrt{3} (- \cos (\frac{π}{3}))$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 4

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$x = 6 \sqrt{3} (- \frac{1}{2})$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = 6 \sqrt{3} (\frac{- 1}{2})$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 5.2

Factor $2$ out of $6 \sqrt{3}$ .

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

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 5.3

Cancel the common factor.

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

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 5.4

Rewrite the expression.

$x = 3 \sqrt{3} \cdot - 1$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

$x = 3 \sqrt{3} \cdot - 1$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 6

Multiply $- 1$ by $3$ .

$x = - 3 \sqrt{3}$

$y = (6 \sqrt{3}) \sin (\frac{4 π}{3})$

Step 7

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.

$x = - 3 \sqrt{3}$

$y = 6 \sqrt{3} (- \sin (\frac{π}{3}))$

Step 8

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$x = - 3 \sqrt{3}$

$y = 6 \sqrt{3} (- \frac{\sqrt{3}}{2})$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$x = - 3 \sqrt{3}$

$y = 6 \sqrt{3} (\frac{- \sqrt{3}}{2})$

Step 9.2

Factor $2$ out of $6 \sqrt{3}$ .

$x = - 3 \sqrt{3}$

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

Step 9.3

Cancel the common factor.

$x = - 3 \sqrt{3}$

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

Step 9.4

Rewrite the expression.

$x = - 3 \sqrt{3}$

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

$x = - 3 \sqrt{3}$

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

Step 10

Multiply $- 1$ by $3$ .

$x = - 3 \sqrt{3}$

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

Step 11

Raise $\sqrt{3}$ to the power of $1$ .

$x = - 3 \sqrt{3}$

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

Step 12

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

$x = - 3 \sqrt{3}$

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

Step 13

Add $1$ and $1$ .

$x = - 3 \sqrt{3}$

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

Step 14

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = - 3 \sqrt{3}$

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

Step 14.2

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

$x = - 3 \sqrt{3}$

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

Step 14.3

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

$x = - 3 \sqrt{3}$

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

Step 14.4

Cancel the common factor of $2$ .

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Step 14.4.1

Cancel the common factor.

$x = - 3 \sqrt{3}$

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

Step 14.4.2

Rewrite the expression.

$x = - 3 \sqrt{3}$

$y = - 3 \cdot 3$

$x = - 3 \sqrt{3}$

$y = - 3 \cdot 3$

Step 14.5

Evaluate the exponent.

$x = - 3 \sqrt{3}$

$y = - 3 \cdot 3$

$x = - 3 \sqrt{3}$

$y = - 3 \cdot 3$

Step 15

Multiply $- 3$ by $3$ .

$x = - 3 \sqrt{3}$

$y = - 9$

Step 16

The rectangular representation of the polar point $(6 \sqrt{3}, \frac{4 π}{3})$ is $(- 3 \sqrt{3}, - 9)$ .

$(- 3 \sqrt{3}, - 9)$