Trigonometry Examples

$A = \frac{1}{2} r^{2} θ$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (r^{2} θ) = A$ .

$\frac{1}{2} \cdot (r^{2} θ) = A$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (r^{2} θ)) = 2 A$

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 (\frac{1}{2} \cdot (r^{2} θ))$ .

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Step 3.1.1

Multiply $\frac{1}{2} (r^{2} θ)$ .

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Step 3.1.1.1

Combine $r^{2}$ and $\frac{1}{2}$ .

$2 (\frac{r^{2}}{2} θ) = 2 A$

Step 3.1.1.2

Combine $\frac{r^{2}}{2}$ and $θ$ .

$2 \frac{r^{2} θ}{2} = 2 A$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

$2 \frac{r^{2} θ}{2} = 2 A$

Step 3.1.2.2

Rewrite the expression.

$r^{2} θ = 2 A$

Step 4

Divide each term in $r^{2} θ = 2 A$ by $θ$ and simplify.

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Step 4.1

Divide each term in $r^{2} θ = 2 A$ by $θ$ .

$\frac{r^{2} θ}{θ} = \frac{2 A}{θ}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $θ$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{r^{2} θ}{θ} = \frac{2 A}{θ}$

Step 4.2.1.2

Divide $r^{2}$ by $1$ .

$r^{2} = \frac{2 A}{θ}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$r = \pm \sqrt{\frac{2 A}{θ}}$

Step 6

Simplify $\pm \sqrt{\frac{2 A}{θ}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{2 A}{θ}}$ as $\frac{\sqrt{2 A}}{\sqrt{θ}}$ .

$r = \pm \frac{\sqrt{2 A}}{\sqrt{θ}}$

Step 6.2

Multiply $\frac{\sqrt{2 A}}{\sqrt{θ}}$ by $\frac{\sqrt{θ}}{\sqrt{θ}}$ .

$r = \pm \frac{\sqrt{2 A}}{\sqrt{θ}} \cdot \frac{\sqrt{θ}}{\sqrt{θ}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{\sqrt{2 A}}{\sqrt{θ}}$ by $\frac{\sqrt{θ}}{\sqrt{θ}}$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{\sqrt{θ} \sqrt{θ}}$

Step 6.3.2

Raise $\sqrt{θ}$ to the power of $1$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{{\sqrt{θ}}^{1} \sqrt{θ}}$

Step 6.3.3

Raise $\sqrt{θ}$ to the power of $1$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{{\sqrt{θ}}^{1} {\sqrt{θ}}^{1}}$

Step 6.3.4

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

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{{\sqrt{θ}}^{1 + 1}}$

Step 6.3.5

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{{\sqrt{θ}}^{2}}$

Step 6.3.6

Rewrite ${\sqrt{θ}}^{2}$ as $θ$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{θ}$ as $θ^{\frac{1}{2}}$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{{(θ^{\frac{1}{2}})}^{2}}$

Step 6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{θ^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{θ^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{θ^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{θ^{1}}$

Step 6.3.6.5

Simplify.

$r = \pm \frac{\sqrt{2 A} \sqrt{θ}}{θ}$

Step 6.4

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{2 A θ}}{θ}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$r = \frac{\sqrt{2 A θ}}{θ}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - \frac{\sqrt{2 A θ}}{θ}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = \frac{\sqrt{2 A θ}}{θ}$

$r = - \frac{\sqrt{2 A θ}}{θ}$

$r = \frac{\sqrt{2 A θ}}{θ}$

$r = - \frac{\sqrt{2 A θ}}{θ}$