Trigonometry Examples

csc (\frac{4 π}{3})

$\csc (\frac{4 π}{3})$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the third 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

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

$3$ .

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Step 4.6.1

Use

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

$\sqrt{3}$ as

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

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

Step 4.6.2

Apply the power rule and multiply exponents,

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

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

Step 4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 4.6.4

Cancel the common factor of

$2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.15470053 \dots$