Trigonometry Examples

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

Step 1

Find $c$ .

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Step 1.1

The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

$\cos (B) = \frac{adj}{hyp}$

Step 1.2

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

$\cos (B) = \frac{a}{c}$

Step 1.3

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

$c = \frac{a}{\cos (B)}$

Step 1.4

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

$c = \frac{5}{\cos (45)}$

Step 1.5

The value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.6

Multiply the numerator by the reciprocal of the denominator.

$c = 5 (\frac{2}{\sqrt{2}})$

Step 1.7

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

$c = 5 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.8

Combine and simplify the denominator.

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Step 1.8.1

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

$c = 5 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.2

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

$c = 5 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.3

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

$c = 5 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.4

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

$c = 5 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Step 1.8.5

Add $1$ and $1$ .

$c = 5 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 1.8.6

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

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Step 1.8.6.1

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

$c = 5 (\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 1.8.6.2

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

$c = 5 (\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 1.8.6.3

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

$c = 5 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.8.6.4

Cancel the common factor of $2$ .

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Step 1.8.6.4.1

Cancel the common factor.

$c = 5 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.8.6.4.2

Rewrite the expression.

$c = 5 (\frac{2 \sqrt{2}}{2})$

Step 1.8.6.5

Evaluate the exponent.

$c = 5 (\frac{2 \sqrt{2}}{2})$

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

Cancel the common factor.

$c = 5 (\frac{2 \sqrt{2}}{2})$

Step 1.9.2

Divide $\sqrt{2}$ by $1$ .

$c = 5 \sqrt{2}$

Step 2

Find the last side of the triangle using the Pythagorean theorem.

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Step 2.1

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

$a^{2} + b^{2} = c^{2}$

Step 2.2

Solve the equation for $b$ .

$b = \sqrt{c^{2} - a^{2}}$

Step 2.3

Substitute the actual values into the equation.

$b = \sqrt{{(5 \sqrt{2})}^{2} - {(5)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Apply the product rule to $5 \sqrt{2}$ .

$b = \sqrt{5^{2} {\sqrt{2}}^{2} - {(5)}^{2}}$

Step 2.4.2

Raise $5$ to the power of $2$ .

$b = \sqrt{25 {\sqrt{2}}^{2} - {(5)}^{2}}$

Step 2.5

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

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Step 2.5.1

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

$b = \sqrt{25 {(2^{\frac{1}{2}})}^{2} - {(5)}^{2}}$

Step 2.5.2

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

$b = \sqrt{25 \cdot 2^{\frac{1}{2} \cdot 2} - {(5)}^{2}}$

Step 2.5.3

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

$b = \sqrt{25 \cdot 2^{\frac{2}{2}} - {(5)}^{2}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$b = \sqrt{25 \cdot 2^{\frac{2}{2}} - {(5)}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$b = \sqrt{25 \cdot 2 - {(5)}^{2}}$

Step 2.5.5

Evaluate the exponent.

$b = \sqrt{25 \cdot 2 - {(5)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $25$ by $2$ .

$b = \sqrt{50 - {(5)}^{2}}$

Step 2.6.2

Raise $5$ to the power of $2$ .

$b = \sqrt{50 - 1 \cdot 25}$

Step 2.6.3

Multiply $- 1$ by $25$ .

$b = \sqrt{50 - 25}$

Step 2.6.4

Subtract $25$ from $50$ .

$b = \sqrt{25}$

Step 2.6.5

Rewrite $25$ as $5^{2}$ .

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

Step 2.7

Pull terms out from under the radical, assuming positive real numbers.

$b = 5$

Step 3

These are the results for all angles and sides for the given triangle.

$A = 45$

$B = 45$

$C = 90$

$a = 5$

$b = 5$

$c = 5 \sqrt{2}$