Find the Length of b tri{}{30}{4}{60}{}{90}

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 4 \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

The sine of an angle is equal to the ratio of the opposite side to the hypotenuse.

$\sin (B) = \frac{opp}{hyp}$

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

$\sin (B) = \frac{b}{c}$

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

$b = c \cdot \sin (B)$

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

$b = c \cdot \sin (60)$

Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

$b = 2 (2) \cdot \frac{\sqrt{3}}{2}$

Cancel the common factor.

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

Rewrite the expression.

$b = 2 \cdot \sqrt{3}$

$b = 2 \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$b = 2 \sqrt{3}$

Decimal Form:

$b = 3.46410161 \dots$

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