Trigonometry Examples

$(5 \sqrt{3}, π)$

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

$x = (5 \sqrt{3}) \cos (π)$

$y = (5 \sqrt{3}) \sin (π)$

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 second quadrant.

$x = 5 \sqrt{3} (- \cos (0))$

$y = (5 \sqrt{3}) \sin (π)$

Step 4

The exact value of $\cos (0)$ is $1$ .

$x = 5 \sqrt{3} (- 1 \cdot 1)$

$y = (5 \sqrt{3}) \sin (π)$

Step 5

Multiply $- 1$ by $1$ .

$x = 5 \sqrt{3} \cdot - 1$

$y = (5 \sqrt{3}) \sin (π)$

Step 6

Multiply $- 1$ by $5$ .

$x = - 5 \sqrt{3}$

$y = (5 \sqrt{3}) \sin (π)$

Step 7

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

$x = - 5 \sqrt{3}$

$y = 5 \sqrt{3} \sin (0)$

Step 8

The exact value of $\sin (0)$ is $0$ .

$x = - 5 \sqrt{3}$

$y = 5 \sqrt{3} \cdot 0$

Step 9

Multiply $5 \sqrt{3} \cdot 0$ .

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Step 9.1

Multiply $0$ by $5$ .

$x = - 5 \sqrt{3}$

$y = 0 \sqrt{3}$

Step 9.2

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

$x = - 5 \sqrt{3}$

$y = 0$

$x = - 5 \sqrt{3}$

$y = 0$

Step 10

The rectangular representation of the polar point $(5 \sqrt{3}, π)$ is $(- 5 \sqrt{3}, 0)$ .

$(- 5 \sqrt{3}, 0)$