Trigonometry Examples

$\csc (\frac{7 π}{3})$

Step 1

Find the value using the definition of cosecant.

$\csc (\frac{7 π}{3}) = \frac{hypotenuse}{opposite}$

Step 2

Substitute the values into the definition.

$\csc (\frac{7 π}{3}) = \frac{1}{\frac{\sqrt{3}}{2}}$

Step 3

Simplify the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Multiply $\frac{2}{\sqrt{3}}$ by $1$ .

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

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.4.6

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

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Step 3.4.6.1

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 3.4.6.2

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

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

Step 3.4.6.3

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

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

Step 3.4.6.4

Cancel the common factor of $2$ .

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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.

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

Step 3.4.6.5

Evaluate the exponent.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.15470053 \dots$

Step 5