Convert to Rectangular r=8sec(theta)

$r = 8 \sec (θ)$

Step 1

Simplify the right side.

Simplify $8 \sec (θ)$ .

Rewrite $\sec (θ)$ in terms of sines and cosines.

$r = 8 \frac{1}{\cos (θ)}$

Combine $8$ and $\frac{1}{\cos (θ)}$ .

$r = \frac{8}{\cos (θ)}$

Step 2

Since $\cos (θ) = \frac{x}{r}$ , replace $\cos (θ)$ with $\frac{x}{r}$ .

$r = \frac{8}{\frac{x}{r}}$

Step 3

Multiply both sides by $r$ .

$r \cdot r = r \frac{8}{\frac{x}{r}}$

Simplify the left side.

Multiply $r$ by $r$ .

$r^{2} = r \frac{8}{\frac{x}{r}}$

Simplify the right side.

Simplify $r \frac{8}{\frac{x}{r}}$ .

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = r (8 \frac{r}{x})$

Rewrite using the commutative property of multiplication.

$r^{2} = 8 r \frac{r}{x}$

Multiply $8 r \frac{r}{x}$ .

Combine $8$ and $\frac{r}{x}$ .

$r^{2} = r \frac{8 r}{x}$

Combine $r$ and $\frac{8 r}{x}$ .

$r^{2} = \frac{r (8 r)}{x}$

Raise $r$ to the power of $1$ .

$r^{2} = \frac{8 (r^{1} r)}{x}$

Raise $r$ to the power of $1$ .

$r^{2} = \frac{8 (r^{1} r^{1})}{x}$

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

$r^{2} = \frac{8 r^{1 + 1}}{x}$

Add $1$ and $1$ .

$r^{2} = \frac{8 r^{2}}{x}$

Step 4

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = \frac{8 (x^{2} + y^{2})}{x}$