Trigonometry Examples

$\csc (60) = \frac{26}{h}$

Step 1

Rewrite the equation as $\frac{26}{h} = \csc (60)$ .

$\frac{26}{h} = \csc (60)$

Step 2

Simplify both sides.

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Step 2.1

The exact value of $\csc (60)$ is $\frac{2}{\sqrt{3}}$ .

$\frac{26}{h} = \frac{2}{\sqrt{3}}$

Step 2.2

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{26}{h} = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{26}{h} = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{26}{h} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{26}{h} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.3.4

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

$\frac{26}{h} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{26}{h} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{26}{h} = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.3.6.2

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

$\frac{26}{h} = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.3.6.3

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

$\frac{26}{h} = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

$\frac{26}{h} = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.3.6.4.2

Rewrite the expression.

$\frac{26}{h} = \frac{2 \sqrt{3}}{3^{1}}$

Step 2.3.6.5

Evaluate the exponent.

$\frac{26}{h} = \frac{2 \sqrt{3}}{3}$

Step 3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$26 \cdot 3 = h (2 \sqrt{3})$

Step 4

Solve the equation for $h$ .

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Step 4.1

Rewrite the equation as $h \cdot (2 \sqrt{3}) = 26 \cdot 3$ .

$h \cdot (2 \sqrt{3}) = 26 \cdot 3$

Step 4.2

Multiply $26$ by $3$ .

$h \cdot 2 \sqrt{3} = 78$

Step 4.3

Divide each term in $h \cdot (2 \sqrt{3}) = 78$ by $2 \sqrt{3}$ and simplify.

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Step 4.3.1

Divide each term in $h \cdot (2 \sqrt{3}) = 78$ by $2 \sqrt{3}$ .

$\frac{h \cdot (2 \sqrt{3})}{2 \sqrt{3}} = \frac{78}{2 \sqrt{3}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{h \cdot (2 \sqrt{3})}{2 \sqrt{3}} = \frac{78}{2 \sqrt{3}}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{h \cdot (\sqrt{3})}{\sqrt{3}} = \frac{78}{2 \sqrt{3}}$

Step 4.3.2.2

Cancel the common factor of $\sqrt{3}$ .

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Step 4.3.2.2.1

Cancel the common factor.

$\frac{h \cdot \sqrt{3}}{\sqrt{3}} = \frac{78}{2 \sqrt{3}}$

Step 4.3.2.2.2

Divide $h$ by $1$ .

$h = \frac{78}{2 \sqrt{3}}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Cancel the common factor of $78$ and $2$ .

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Step 4.3.3.1.1

Factor $2$ out of $78$ .

$h = \frac{2 \cdot 39}{2 \sqrt{3}}$

Step 4.3.3.1.2

Cancel the common factors.

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Step 4.3.3.1.2.1

Factor $2$ out of $2 \sqrt{3}$ .

$h = \frac{2 \cdot 39}{2 (\sqrt{3})}$

Step 4.3.3.1.2.2

Cancel the common factor.

$h = \frac{2 \cdot 39}{2 \sqrt{3}}$

Step 4.3.3.1.2.3

Rewrite the expression.

$h = \frac{39}{\sqrt{3}}$

Step 4.3.3.2

Multiply $\frac{39}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$h = \frac{39}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.3.3.3

Combine and simplify the denominator.

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Step 4.3.3.3.1

Multiply $\frac{39}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$h = \frac{39 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.3.3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$h = \frac{39 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.3.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$h = \frac{39 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.3.3.3.4

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

$h = \frac{39 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.3.3.3.5

Add $1$ and $1$ .

$h = \frac{39 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.3.3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$h = \frac{39 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.3.3.3.6.2

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

$h = \frac{39 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.3.3.3.6.3

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

$h = \frac{39 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.3.3.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.3.3.6.4.1

Cancel the common factor.

$h = \frac{39 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.3.3.3.6.4.2

Rewrite the expression.

$h = \frac{39 \sqrt{3}}{3^{1}}$

Step 4.3.3.3.6.5

Evaluate the exponent.

$h = \frac{39 \sqrt{3}}{3}$

Step 4.3.3.4

Cancel the common factor of $39$ and $3$ .

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Step 4.3.3.4.1

Factor $3$ out of $39 \sqrt{3}$ .

$h = \frac{3 (13 \sqrt{3})}{3}$

Step 4.3.3.4.2

Cancel the common factors.

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Step 4.3.3.4.2.1

Factor $3$ out of $3$ .

$h = \frac{3 (13 \sqrt{3})}{3 (1)}$

Step 4.3.3.4.2.2

Cancel the common factor.

$h = \frac{3 (13 \sqrt{3})}{3 \cdot 1}$

Step 4.3.3.4.2.3

Rewrite the expression.

$h = \frac{13 \sqrt{3}}{1}$

Step 4.3.3.4.2.4

Divide $13 \sqrt{3}$ by $1$ .

$h = 13 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$h = 13 \sqrt{3}$

Decimal Form:

$h = 22.51666049 \dots$