Trigonometry Examples

$(3 \sqrt{3}, \frac{3 π}{2})$

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

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

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

Step 3

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

$x = 3 \sqrt{3} \cos (\frac{π}{2})$

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

Step 4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$x = 3 \sqrt{3} \cdot 0$

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

Step 5

Multiply $3 \sqrt{3} \cdot 0$ .

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Step 5.1

Multiply $0$ by $3$ .

$x = 0 \sqrt{3}$

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

Step 5.2

Multiply $0$ by $\sqrt{3}$ .

$x = 0$

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

$x = 0$

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

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.

$x = 0$

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

Step 7

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

$x = 0$

$y = 3 \sqrt{3} (- 1 \cdot 1)$

Step 8

Multiply $- 1$ by $1$ .

$x = 0$

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

Step 9

Multiply $- 1$ by $3$ .

$x = 0$

$y = - 3 \sqrt{3}$

Step 10

The rectangular representation of the polar point $(3 \sqrt{3}, \frac{3 π}{2})$ is $(0, - 3 \sqrt{3})$ .

$(0, - 3 \sqrt{3})$