Find the Length of a tri{}{45}{6}{45}{}{90}

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 6 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

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

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

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

$\sin (A) = \frac{a}{c}$

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

$a = c \cdot \sin (A)$

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

$a = c \cdot \sin (45)$

Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

$a = 3 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$a = 3 \sqrt{2}$

Decimal Form:

$a = 4.24264068 \dots$

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