Trigonometry Examples

$c = 2 π \sqrt{\frac{a^{2} + b^{2}}{2}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{a^{2} + b^{2}}{2}} = c$ .

$2 π \sqrt{\frac{a^{2} + b^{2}}{2}} = c$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 π \sqrt{\frac{a^{2} + b^{2}}{2}})}^{2} = c^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(2 π {(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}})}^{2} = c^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(2 π {(\frac{a^{2} + b^{2}}{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $\frac{a^{2} + b^{2}}{2}$ .

${(2 π \frac{{(a^{2} + b^{2})}^{\frac{1}{2}}}{2^{\frac{1}{2}}})}^{2} = c^{2}$

Step 3.2.1.2

Multiply $2 π \frac{{(a^{2} + b^{2})}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ .

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Step 3.2.1.2.1

Combine $\frac{{(a^{2} + b^{2})}^{\frac{1}{2}}}{2^{\frac{1}{2}}}$ and $2$ .

${(\frac{{(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2}{2^{\frac{1}{2}}} π)}^{2} = c^{2}$

Step 3.2.1.2.2

Combine $\frac{{(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2}{2^{\frac{1}{2}}}$ and $π$ .

${(\frac{{(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2 π}{2^{\frac{1}{2}}})}^{2} = c^{2}$

Step 3.2.1.3

Move $2^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2 π \cdot 2^{- \frac{1}{2}})}^{2} = c^{2}$

Step 3.2.1.4

Multiply $2$ by $2^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.2.1.4.1

Move $2^{- \frac{1}{2}}$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot (2^{- \frac{1}{2}} \cdot 2) π)}^{2} = c^{2}$

Step 3.2.1.4.2

Multiply $2^{- \frac{1}{2}}$ by $2$ .

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Step 3.2.1.4.2.1

Raise $2$ to the power of $1$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot (2^{- \frac{1}{2}} \cdot 2^{1}) π)}^{2} = c^{2}$

Step 3.2.1.4.2.2

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

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{- \frac{1}{2} + 1} π)}^{2} = c^{2}$

Step 3.2.1.4.3

Write $1$ as a fraction with a common denominator.

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{- \frac{1}{2} + \frac{2}{2}} π)}^{2} = c^{2}$

Step 3.2.1.4.4

Combine the numerators over the common denominator.

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{\frac{- 1 + 2}{2}} π)}^{2} = c^{2}$

Step 3.2.1.4.5

Add $- 1$ and $2$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} π)}^{2} = c^{2}$

Step 3.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.5.1

Apply the product rule to ${(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} π$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.5.2

Apply the product rule to ${(a^{2} + b^{2})}^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}$ .

${({(a^{2} + b^{2})}^{\frac{1}{2}})}^{2} \cdot {(2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.6

Multiply the exponents in ${({(a^{2} + b^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.6.1

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

${(a^{2} + b^{2})}^{\frac{1}{2} \cdot 2} \cdot {(2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.6.2

Cancel the common factor of $2$ .

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Step 3.2.1.6.2.1

Cancel the common factor.

${(a^{2} + b^{2})}^{\frac{1}{2} \cdot 2} \cdot {(2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.6.2.2

Rewrite the expression.

${(a^{2} + b^{2})}^{1} \cdot {(2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.7

Simplify.

$(a^{2} + b^{2}) \cdot {(2^{\frac{1}{2}})}^{2} π^{2} = c^{2}$

Step 3.2.1.8

Multiply the exponents in ${(2^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.8.1

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

$(a^{2} + b^{2}) \cdot 2^{\frac{1}{2} \cdot 2} π^{2} = c^{2}$

Step 3.2.1.8.2

Cancel the common factor of $2$ .

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Step 3.2.1.8.2.1

Cancel the common factor.

$(a^{2} + b^{2}) \cdot 2^{\frac{1}{2} \cdot 2} π^{2} = c^{2}$

Step 3.2.1.8.2.2

Rewrite the expression.

$(a^{2} + b^{2}) \cdot 2^{1} π^{2} = c^{2}$

Step 3.2.1.9

Evaluate the exponent.

$(a^{2} + b^{2}) \cdot 2 π^{2} = c^{2}$

Step 3.2.1.10

Simplify by multiplying through.

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Step 3.2.1.10.1

Apply the distributive property.

$(a^{2} \cdot 2 + b^{2} \cdot 2) π^{2} = c^{2}$

Step 3.2.1.10.2

Reorder.

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Step 3.2.1.10.2.1

Move $2$ to the left of $a^{2}$ .

$(2 \cdot a^{2} + b^{2} \cdot 2) π^{2} = c^{2}$

Step 3.2.1.10.2.2

Move $2$ to the left of $b^{2}$ .

$(2 \cdot a^{2} + 2 \cdot b^{2}) π^{2} = c^{2}$

Step 3.2.1.11

Multiply $2$ by $b^{2}$ .

$(2 a^{2} + 2 b^{2}) π^{2} = c^{2}$

Step 3.2.1.12

Apply the distributive property.

$2 a^{2} π^{2} + 2 b^{2} π^{2} = c^{2}$

Step 4

Solve for $b$ .

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Step 4.1

Subtract $2 a^{2} π^{2}$ from both sides of the equation.

$2 b^{2} π^{2} = c^{2} - 2 a^{2} π^{2}$

Step 4.2

Divide each term in $2 b^{2} π^{2} = c^{2} - 2 a^{2} π^{2}$ by $2 π^{2}$ and simplify.

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Step 4.2.1

Divide each term in $2 b^{2} π^{2} = c^{2} - 2 a^{2} π^{2}$ by $2 π^{2}$ .

$\frac{2 b^{2} π^{2}}{2 π^{2}} = \frac{c^{2}}{2 π^{2}} + \frac{- 2 a^{2} π^{2}}{2 π^{2}}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 b^{2} π^{2}}{2 π^{2}} = \frac{c^{2}}{2 π^{2}} + \frac{- 2 a^{2} π^{2}}{2 π^{2}}$

Step 4.2.2.1.2

Rewrite the expression.

$\frac{b^{2} π^{2}}{π^{2}} = \frac{c^{2}}{2 π^{2}} + \frac{- 2 a^{2} π^{2}}{2 π^{2}}$

Step 4.2.2.2

Cancel the common factor of $π^{2}$ .

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Step 4.2.2.2.1

Cancel the common factor.

$\frac{b^{2} π^{2}}{π^{2}} = \frac{c^{2}}{2 π^{2}} + \frac{- 2 a^{2} π^{2}}{2 π^{2}}$

Step 4.2.2.2.2

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

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{- 2 a^{2} π^{2}}{2 π^{2}}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.3.1.1.1

Factor $2$ out of $- 2 a^{2} π^{2}$ .

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{2 (- a^{2} π^{2})}{2 π^{2}}$

Step 4.2.3.1.1.2

Cancel the common factors.

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Step 4.2.3.1.1.2.1

Factor $2$ out of $2 π^{2}$ .

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{2 (- a^{2} π^{2})}{2 (π^{2})}$

Step 4.2.3.1.1.2.2

Cancel the common factor.

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{2 (- a^{2} π^{2})}{2 π^{2}}$

Step 4.2.3.1.1.2.3

Rewrite the expression.

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{- a^{2} π^{2}}{π^{2}}$

Step 4.2.3.1.2

Cancel the common factor of $π^{2}$ .

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Step 4.2.3.1.2.1

Cancel the common factor.

$b^{2} = \frac{c^{2}}{2 π^{2}} + \frac{- a^{2} π^{2}}{π^{2}}$

Step 4.2.3.1.2.2

Divide $- a^{2}$ by $1$ .

$b^{2} = \frac{c^{2}}{2 π^{2}} - a^{2}$

Step 4.3

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

$b = \pm \sqrt{\frac{c^{2}}{2 π^{2}} - a^{2}}$

Step 4.4

Simplify $\pm \sqrt{\frac{c^{2}}{2 π^{2}} - a^{2}}$ .

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Step 4.4.1

To write $- a^{2}$ as a fraction with a common denominator, multiply by $\frac{2 π^{2}}{2 π^{2}}$ .

$b = \pm \sqrt{\frac{c^{2}}{2 π^{2}} - a^{2} \cdot \frac{2 π^{2}}{2 π^{2}}}$

Step 4.4.2

Combine $- a^{2}$ and $\frac{2 π^{2}}{2 π^{2}}$ .

$b = \pm \sqrt{\frac{c^{2}}{2 π^{2}} + \frac{- a^{2} (2 π^{2})}{2 π^{2}}}$

Step 4.4.3

Combine the numerators over the common denominator.

$b = \pm \sqrt{\frac{c^{2} - a^{2} (2 π^{2})}{2 π^{2}}}$

Step 4.4.4

Multiply $2$ by $- 1$ .

$b = \pm \sqrt{\frac{c^{2} - 2 a^{2} π^{2}}{2 π^{2}}}$

Step 4.4.5

Rewrite $\frac{c^{2} - 2 a^{2} π^{2}}{2 π^{2}}$ as ${(\frac{1}{π})}^{2} \frac{c^{2} - 2 a^{2} π^{2}}{2}$ .

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Step 4.4.5.1

Factor the perfect power $1^{2}$ out of $c^{2} - 2 a^{2} π^{2}$ .

$b = \pm \sqrt{\frac{1^{2} (c^{2} - 2 a^{2} π^{2})}{2 π^{2}}}$

Step 4.4.5.2

Factor the perfect power $π^{2}$ out of $2 π^{2}$ .

$b = \pm \sqrt{\frac{1^{2} (c^{2} - 2 a^{2} π^{2})}{π^{2} \cdot 2}}$

Step 4.4.5.3

Rearrange the fraction $\frac{1^{2} (c^{2} - 2 a^{2} π^{2})}{π^{2} \cdot 2}$ .

$b = \pm \sqrt{{(\frac{1}{π})}^{2} \frac{c^{2} - 2 a^{2} π^{2}}{2}}$

Step 4.4.6

Pull terms out from under the radical.

$b = \pm \frac{1}{π} \sqrt{\frac{c^{2} - 2 a^{2} π^{2}}{2}}$

Step 4.4.7

Rewrite $\sqrt{\frac{c^{2} - 2 a^{2} π^{2}}{2}}$ as $\frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{\sqrt{2}}$ .

$b = \pm \frac{1}{π} \cdot \frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{\sqrt{2}}$

Step 4.4.8

Combine.

$b = \pm \frac{1 \sqrt{c^{2} - 2 a^{2} π^{2}}}{π \sqrt{2}}$

Step 4.4.9

Multiply $\sqrt{c^{2} - 2 a^{2} π^{2}}$ by $1$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{π \sqrt{2}}$

Step 4.4.10

Multiply $\frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{π \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{π \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.4.11

Combine and simplify the denominator.

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Step 4.4.11.1

Multiply $\frac{\sqrt{c^{2} - 2 a^{2} π^{2}}}{π \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \sqrt{2} \sqrt{2}}$

Step 4.4.11.2

Move $\sqrt{2}$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π (\sqrt{2} \sqrt{2})}$

Step 4.4.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π ({\sqrt{2}}^{1} \sqrt{2})}$

Step 4.4.11.4

Raise $\sqrt{2}$ to the power of $1$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 4.4.11.5

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

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π {\sqrt{2}}^{1 + 1}}$

Step 4.4.11.6

Add $1$ and $1$ .

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π {\sqrt{2}}^{2}}$

Step 4.4.11.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.4.11.7.1

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

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π {(2^{\frac{1}{2}})}^{2}}$

Step 4.4.11.7.2

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

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.4.11.7.3

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

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \cdot 2^{\frac{2}{2}}}$

Step 4.4.11.7.4

Cancel the common factor of $2$ .

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Step 4.4.11.7.4.1

Cancel the common factor.

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \cdot 2^{\frac{2}{2}}}$

Step 4.4.11.7.4.2

Rewrite the expression.

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \cdot 2^{1}}$

Step 4.4.11.7.5

Evaluate the exponent.

$b = \pm \frac{\sqrt{c^{2} - 2 a^{2} π^{2}} \sqrt{2}}{π \cdot 2}$

Step 4.4.12

Combine using the product rule for radicals.

$b = \pm \frac{\sqrt{(c^{2} - 2 a^{2} π^{2}) \cdot 2}}{π \cdot 2}$

Step 4.4.13

Simplify the expression.

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Step 4.4.13.1

Move $2$ to the left of $π$ .

$b = \pm \frac{\sqrt{(c^{2} - 2 a^{2} π^{2}) \cdot 2}}{2 \cdot π}$

Step 4.4.13.2

Reorder factors in $\pm \frac{\sqrt{(c^{2} - 2 a^{2} π^{2}) \cdot 2}}{2 π}$ .

$b = \pm \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

Step 4.5

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

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Step 4.5.1

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

$b = \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

Step 4.5.2

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

$b = - \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

Step 4.5.3

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

$b = \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

$b = - \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

$b = \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

$b = - \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

$b = \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$

$b = - \frac{\sqrt{2 (c^{2} - 2 a^{2} π^{2})}}{2 π}$