Trigonometry Examples

$T = 2 π \sqrt{\frac{L}{g}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{L}{g}} = T$ .

$2 π \sqrt{\frac{L}{g}} = T$

Step 2

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

${(2 π \sqrt{\frac{L}{g}})}^{2} = T^{2}$

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 $\sqrt{\frac{L}{g}}$ as ${(\frac{L}{g})}^{\frac{1}{2}}$ .

${(2 π {(\frac{L}{g})}^{\frac{1}{2}})}^{2} = T^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(2 π {(\frac{L}{g})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $\frac{L}{g}$ .

${(2 π \frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Step 3.2.1.2

Multiply $2 π \frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ .

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Step 3.2.1.2.1

Combine $\frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ and $2$ .

${(\frac{L^{\frac{1}{2}} \cdot 2}{g^{\frac{1}{2}}} π)}^{2} = T^{2}$

Step 3.2.1.2.2

Combine $\frac{L^{\frac{1}{2}} \cdot 2}{g^{\frac{1}{2}}}$ and $π$ .

${(\frac{L^{\frac{1}{2}} \cdot 2 π}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Step 3.2.1.3

Move $2$ to the left of $L^{\frac{1}{2}}$ .

${(\frac{2 \cdot L^{\frac{1}{2}} π}{g^{\frac{1}{2}}})}^{2} = T^{2}$

Step 3.2.1.4

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

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Step 3.2.1.4.1

Apply the product rule to $\frac{2 L^{\frac{1}{2}} π}{g^{\frac{1}{2}}}$ .

$\frac{{(2 L^{\frac{1}{2}} π)}^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4.2

Apply the product rule to $2 L^{\frac{1}{2}} π$ .

$\frac{{(2 L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.4.3

Apply the product rule to $2 L^{\frac{1}{2}}$ .

$\frac{2^{2} {(L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5

Simplify the numerator.

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Step 3.2.1.5.1

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

$\frac{4 {(L^{\frac{1}{2}})}^{2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5.2

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

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Step 3.2.1.5.2.1

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

$\frac{4 L^{\frac{1}{2} \cdot 2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5.2.2

Cancel the common factor of $2$ .

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Step 3.2.1.5.2.2.1

Cancel the common factor.

$\frac{4 L^{\frac{1}{2} \cdot 2} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5.2.2.2

Rewrite the expression.

$\frac{4 L^{1} π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.5.3

Simplify.

$\frac{4 L π^{2}}{{(g^{\frac{1}{2}})}^{2}} = T^{2}$

Step 3.2.1.6

Simplify the denominator.

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Step 3.2.1.6.1

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

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Step 3.2.1.6.1.1

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

$\frac{4 L π^{2}}{g^{\frac{1}{2} \cdot 2}} = T^{2}$

Step 3.2.1.6.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.6.1.2.1

Cancel the common factor.

$\frac{4 L π^{2}}{g^{\frac{1}{2} \cdot 2}} = T^{2}$

Step 3.2.1.6.1.2.2

Rewrite the expression.

$\frac{4 L π^{2}}{g^{1}} = T^{2}$

Step 3.2.1.6.2

Simplify.

$\frac{4 L π^{2}}{g} = T^{2}$

Step 4

Solve for $L$ .

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Step 4.1

Multiply both sides by $g$ .

$\frac{4 L π^{2}}{g} g = T^{2} g$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $g$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 L π^{2}}{g} g = T^{2} g$

Step 4.2.1.2

Rewrite the expression.

$4 L π^{2} = T^{2} g$

Step 4.3

Divide each term in $4 L π^{2} = T^{2} g$ by $4 π^{2}$ and simplify.

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Step 4.3.1

Divide each term in $4 L π^{2} = T^{2} g$ by $4 π^{2}$ .

$\frac{4 L π^{2}}{4 π^{2}} = \frac{T^{2} g}{4 π^{2}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{4 L π^{2}}{4 π^{2}} = \frac{T^{2} g}{4 π^{2}}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{L π^{2}}{π^{2}} = \frac{T^{2} g}{4 π^{2}}$

Step 4.3.2.2

Cancel the common factor of $π^{2}$ .

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Step 4.3.2.2.1

Cancel the common factor.

$\frac{L π^{2}}{π^{2}} = \frac{T^{2} g}{4 π^{2}}$

Step 4.3.2.2.2

Divide $L$ by $1$ .

$L = \frac{T^{2} g}{4 π^{2}}$