Trigonometry Examples

csc (120)

$\csc (120)$

Step 1

Find the value using the definition of cosecant.

csc (120) = \frac{hypotenuse}{opposite}

$\csc (120) = \frac{hypotenuse}{opposite}$

Step 2

Substitute the values into the definition.

csc (120) = \frac{1}{\frac{\sqrt{3}}{2}}

$\csc (120) = \frac{1}{\frac{\sqrt{3}}{2}}$

Step 3

Simplify the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

1 \frac{2}{\sqrt{3}}

$1 \frac{2}{\sqrt{3}}$

Step 3.2

Multiply

\frac{2}{\sqrt{3}}

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

1

$1$ .

\frac{2}{\sqrt{3}}

$\frac{2}{\sqrt{3}}$

Step 3.3

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.5

Add

1

$1$ and

1

$1$ .

\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 3.4.6.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}

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

Step 3.4.6.2

Apply the power rule and multiply exponents,

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

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

\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{2 \sqrt{3}}{3^{\frac{2}{2}}}

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of

2

$2$ .

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Step 3.4.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}}$

$\frac{2 \sqrt{3}}{3^{1}}$

Step 3.4.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3}$

$\frac{2 \sqrt{3}}{3}$

$\frac{2 \sqrt{3}}{3}$

$\frac{2 \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3}$

Decimal Form:

$1.15470053 \dots$

Step 5

$\csc (120)$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$