Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 4 \\ c = \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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

Step 2

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

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

Step 3

Set up the equation to solve for the hypotenuse, in this case $c$ .

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

Step 4

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

$c = \frac{4}{\sin (45)}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$c = 4 (\frac{2}{\sqrt{2}})$

Step 6

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

$c = 4 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

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

$c = 4 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 7.2

Raise $\sqrt{2}$ to the power of $1$ .

$c = 4 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 7.3

Raise $\sqrt{2}$ to the power of $1$ .

$c = 4 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$c = 4 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Step 7.5

Add $1$ and $1$ .

$c = 4 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$c = 4 (\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$c = 4 (\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 7.6.3

Combine $\frac{1}{2}$ and $2$ .

$c = 4 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$c = 4 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 7.6.4.2

Rewrite the expression.

$c = 4 (\frac{2 \sqrt{2}}{2})$

Step 7.6.5

Evaluate the exponent.

$c = 4 (\frac{2 \sqrt{2}}{2})$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4$ .

$c = 2 (2) (\frac{2 \sqrt{2}}{2})$

Step 8.2

Cancel the common factor.

$c = 2 \cdot (2 (\frac{2 \sqrt{2}}{2}))$

Step 8.3

Rewrite the expression.

$c = 2 (2 \sqrt{2})$

Step 9

Multiply $2$ by $2$ .

$c = 4 \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$c = 4 \sqrt{2}$

Decimal Form:

$c = 5.65685424 \dots$