Trigonometry Examples

T = 2 π \sqrt{\frac{L}{g}}

$T = 2 π \sqrt{\frac{L}{g}}$

Step 1

Rewrite the equation as

2 π \sqrt{\frac{L}{g}} = T

$2 π \sqrt{\frac{L}{g}} = T$ .

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}

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{\frac{L}{g}}

$\sqrt{\frac{L}{g}}$ as

{(\frac{L}{g})}^{\frac{1}{2}}

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

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

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

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Step 3.2.1.1

Apply the product rule to

\frac{L}{g}

$\frac{L}{g}$ .

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

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

Step 3.2.1.2

Multiply

2 π \frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}

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

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Step 3.2.1.2.1

Combine

\frac{L^{\frac{1}{2}}}{g^{\frac{1}{2}}}

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

2

$2$ .

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

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

π

$π$ .

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

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

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

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

Step 3.2.1.3

Move

2

$2$ to the left of

L^{\frac{1}{2}}

$L^{\frac{1}{2}}$ .

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

${(a b)}^{n} = a^{n} b^{n}$ 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}}}

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

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

$2 L^{\frac{1}{2}} π$ .

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

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

Step 3.2.1.4.3

Apply the product rule to

2 L^{\frac{1}{2}}

$2 L^{\frac{1}{2}}$ .

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

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

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

$2$ to the power of

2

$2$ .

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

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

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Step 3.2.1.5.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 3.2.1.5.2.2

Cancel the common factor of

2

$2$ .

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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

$1$ .

$L = \frac{T^{2} g}{4 π^{2}}$