Trigonometry Examples

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\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 2 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 2 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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

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

Step 2

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

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

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

Step 3

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

a

$a$ .

a = c \cdot \sin (A)

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

Step 4

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

a = c \cdot \sin (45)

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

Step 5

Cancel the common factor of

2

$2$ .

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Step 5.1

Cancel the common factor.

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

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

Step 5.2

Rewrite the expression.

a = \sqrt{2}

$a = \sqrt{2}$

a = \sqrt{2}

$a = \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

a = \sqrt{2}

$a = \sqrt{2}$

Decimal Form:

a = 1.41421356 \dots

$a = 1.41421356 \dots$