Trigonometry Examples

$(4 \sqrt{3}, \frac{π}{3})$

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

$x = (4 \sqrt{3}) \cos (\frac{π}{3})$

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

Step 3

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

$x = 4 \sqrt{3} (\frac{1}{2})$

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $4 \sqrt{3}$ .

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

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

Step 4.2

Cancel the common factor.

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

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

Step 4.3

Rewrite the expression.

$x = 2 \sqrt{3}$

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

$x = 2 \sqrt{3}$

$y = (4 \sqrt{3}) \sin (\frac{π}{3})$

Step 5

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

$x = 2 \sqrt{3}$

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $4 \sqrt{3}$ .

$x = 2 \sqrt{3}$

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

Step 6.2

Cancel the common factor.

$x = 2 \sqrt{3}$

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

Step 6.3

Rewrite the expression.

$x = 2 \sqrt{3}$

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

$x = 2 \sqrt{3}$

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

Step 7

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

$x = 2 \sqrt{3}$

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

Step 8

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

$x = 2 \sqrt{3}$

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

Step 9

Add $1$ and $1$ .

$x = 2 \sqrt{3}$

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

Step 10

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

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Step 10.1

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

$x = 2 \sqrt{3}$

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

Step 10.2

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

$x = 2 \sqrt{3}$

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

Step 10.3

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

$x = 2 \sqrt{3}$

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

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$x = 2 \sqrt{3}$

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

Step 10.4.2

Rewrite the expression.

$x = 2 \sqrt{3}$

$y = 2 \cdot 3$

$x = 2 \sqrt{3}$

$y = 2 \cdot 3$

Step 10.5

Evaluate the exponent.

$x = 2 \sqrt{3}$

$y = 2 \cdot 3$

$x = 2 \sqrt{3}$

$y = 2 \cdot 3$

Step 11

Multiply $2$ by $3$ .

$x = 2 \sqrt{3}$

$y = 6$

Step 12

The rectangular representation of the polar point $(4 \sqrt{3}, \frac{π}{3})$ is $(2 \sqrt{3}, 6)$ .

$(2 \sqrt{3}, 6)$