Trigonometry Examples

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\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 5 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 5 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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

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

Step 2

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

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

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

Step 3

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

b

$b$ .

b = c \cdot \sin (B)

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

Step 4

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

b = c \cdot \sin (45)

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

Step 5

Combine

5

$5$ and

\frac{\sqrt{2}}{2}

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

b = \frac{5 \sqrt{2}}{2}

$b = \frac{5 \sqrt{2}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

b = \frac{5 \sqrt{2}}{2}

$b = \frac{5 \sqrt{2}}{2}$

Decimal Form:

b = 3.53553390 \dots

$b = 3.53553390 \dots$