Precalculus Examples

$r = 2 \sec (θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $2 \sec (θ)$ .

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Step 1.1.1

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

$r = 2 \frac{1}{\cos (θ)}$

Step 1.1.2

Combine $2$ and $\frac{1}{\cos (θ)}$ .

$r = \frac{2}{\cos (θ)}$

Step 2

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

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

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r \frac{2}{\frac{x}{r}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r \frac{2}{\frac{x}{r}}$ .

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Step 3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.3

Multiply $2 r \frac{r}{x}$ .

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Step 3.3.1.3.1

Combine $2$ and $\frac{r}{x}$ .

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

Step 3.3.1.3.2

Combine $r$ and $\frac{2 r}{x}$ .

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

Step 3.3.1.3.3

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

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

Step 3.3.1.3.4

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

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

Step 3.3.1.3.5

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

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

Step 3.3.1.3.6

Add $1$ and $1$ .

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

Step 4

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

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