Find the Length of a tri{12}{30}{}{60}{}{90}

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 12 \\ c = \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

$\tan (A) = \frac{opp}{adj}$

Substitute the name of each side into the definition of the tangent function.

$\tan (A) = \frac{a}{b}$

Set up the equation to solve for the opposite side, in this case $a$ .

$a = b \cdot \tan (A)$

Substitute the values of each variable into the formula for tangent.

$a = 12 \cdot \tan (30)$

Cancel the common factor of $3$ .

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Factor $3$ out of $12$ .

$a = 3 (4) \cdot \frac{\sqrt{3}}{3}$

Cancel the common factor.

$a = 3 \cdot 4 \cdot \frac{\sqrt{3}}{3}$

Rewrite the expression.

$a = 4 \cdot \sqrt{3}$

$a = 4 \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$a = 4 \sqrt{3}$

Decimal Form:

$a = 6.92820323 \dots$

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