Trigonometry Examples

$\frac{\tan (\frac{π}{3})}{\csc (\frac{5 π}{3})}$

Step 1

Rewrite $\csc (\frac{5 π}{3})$ in terms of sines and cosines.

$\frac{\tan (\frac{π}{3})}{\frac{1}{\sin (\frac{5 π}{3})}}$

Step 2

Rewrite $\tan (\frac{π}{3})$ in terms of sines and cosines.

$\frac{\frac{\sin (\frac{π}{3})}{\cos (\frac{π}{3})}}{\frac{1}{\sin (\frac{5 π}{3})}}$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (\frac{5 π}{3})}$ .

$\frac{\sin (\frac{π}{3}) \sin (\frac{5 π}{3})}{\cos (\frac{π}{3})}$

Step 4

Convert from $\frac{\sin (\frac{π}{3}) \sin (\frac{5 π}{3})}{\cos (\frac{π}{3})}$ to $\sin (\frac{π}{3}) \tan (\frac{π}{3})$ .

$\sin (\frac{π}{3}) \tan (\frac{π}{3})$

Step 5

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

$\frac{\sqrt{3}}{2} \tan (\frac{π}{3})$

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 8

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

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Step 8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

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

Step 8.5

Evaluate the exponent.

$\frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$