Trigonometry Examples

$\csc (4 \frac{2}{3} \cdot π)$

Step 1

Convert $4 \frac{2}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\csc ((4 + \frac{2}{3}) \cdot π)$

Step 1.2

Add $4$ and $\frac{2}{3}$ .

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Step 1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\csc ((4 \cdot \frac{3}{3} + \frac{2}{3}) \cdot π)$

Step 1.2.2

Combine $4$ and $\frac{3}{3}$ .

$\csc ((\frac{4 \cdot 3}{3} + \frac{2}{3}) \cdot π)$

Step 1.2.3

Combine the numerators over the common denominator.

$\csc (\frac{4 \cdot 3 + 2}{3} \cdot π)$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $4$ by $3$ .

$\csc (\frac{12 + 2}{3} \cdot π)$

Step 1.2.4.2

Add $12$ and $2$ .

$\csc (\frac{14}{3} \cdot π)$

Step 2

Combine $\frac{14}{3}$ and $π$ .

$\csc (\frac{14 π}{3})$

Step 3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\csc (\frac{2 π}{3})$

Step 4

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

$\csc (\frac{π}{3})$

Step 5

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

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.15470053 \dots$