Algebra Examples

$(\sqrt{3}, \frac{5 π}{3})$

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

$x = (\sqrt{3}) \cos (\frac{5 π}{3})$

$y = (\sqrt{3}) \sin (\frac{5 π}{3})$

Step 3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = \sqrt{3} \cos (\frac{π}{3})$

$y = (\sqrt{3}) \sin (\frac{5 π}{3})$

Step 4

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

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

$y = (\sqrt{3}) \sin (\frac{5 π}{3})$

Step 5

Combine $\sqrt{3}$ and $\frac{1}{2}$ .

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

$y = (\sqrt{3}) \sin (\frac{5 π}{3})$

Step 6

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 fourth quadrant.

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

$y = \sqrt{3} (- \sin (\frac{π}{3}))$

Step 7

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

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

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

Step 8

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

Tap for more steps...

Step 8.1

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

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

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

Step 8.2

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

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

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

Step 8.3

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

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

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

Step 8.4

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

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

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

Step 8.5

Add $1$ and $1$ .

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

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

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

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

Step 9

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

Tap for more steps...

Step 9.1

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

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

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

Step 9.2

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

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

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

Step 9.3

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

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

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

Step 9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.4.1

Cancel the common factor.

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

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

Step 9.4.2

Rewrite the expression.

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

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

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

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

Step 9.5

Evaluate the exponent.

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

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

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

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

Step 10

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

$(\frac{\sqrt{3}}{2}, - \frac{3}{2})$