Trigonometry Examples

\csc (\frac{2 π}{3})

$\csc (\frac{2 π}{3})$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

\csc (\frac{π}{3})

$\csc (\frac{π}{3})$

Step 2

The exact value of

\csc (\frac{π}{3})

$\csc (\frac{π}{3})$ is

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

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

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

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

Step 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 4

Combine and simplify the denominator.

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Step 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 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 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 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 4.5

Add

1

$1$ and

1

$1$ .

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

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

Step 4.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 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 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 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 4.6.4

Cancel the common factor of

2

$2$ .

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Step 4.6.4.1

Cancel the common factor.

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

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

Step 4.6.4.2

Rewrite the expression.

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

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

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

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

Step 4.6.5

Evaluate the exponent.

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

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

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

1.15470053 \dots

$1.15470053 \dots$