Calculus Examples

$(5, π)$

Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

$x = r c o s θ$

$y = r s i n θ$

Substitute in the known values of $r = 5$ and $θ = π$ into the formulas.

$x = (5) \cos (π)$

$y = (5) \sin (π)$

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.

$x = 5 (- \cos (0))$

$y = (5) \sin (π)$

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

$x = 5 (- 1 \cdot 1)$

$y = (5) \sin (π)$

Multiply $- 1$ by $1$ .

$x = 5 \cdot - 1$

$y = (5) \sin (π)$

Multiply $5$ by $- 1$ .

$x = - 5$

$y = (5) \sin (π)$

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

$x = - 5$

$y = 5 \sin (0)$

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

$x = - 5$

$y = 5 \cdot 0$

Multiply $5$ by $0$ .

$x = - 5$

$y = 0$

The rectangular representation of the polar point $(5, π)$ is $(- 5, 0)$ .

$(- 5, 0)$