Trigonometry Examples

$\tan (30)$

Find the value using the definition of tangent.

$\tan (30) = \frac{opposite}{adjacent}$

Substitute the values into the definition.

$\tan (30) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

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Factor out the greatest common factor $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{1}{\sqrt{3}}$ .

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

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

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

Simplify.

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Combine.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

The result can be shown in both exact and decimal forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$

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