Trigonometry Examples

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$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 9 \sqrt{2} \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find

$b$ .

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Step 1.1

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

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

Step 1.2

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

$\cos (A) = \frac{b}{c}$

Step 1.3

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

$b$ .

$b = c \cdot \cos (A)$

Step 1.4

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

$b = 9 \sqrt{2} \cdot \cos (45)$

Step 1.5

Multiply

$9 \sqrt{2} \frac{\sqrt{2}}{2}$ .

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Step 1.5.1

Combine

$\frac{\sqrt{2}}{2}$ and

$9$ .

$b = \frac{\sqrt{2} \cdot 9}{2} \cdot \sqrt{2}$

Step 1.5.2

Combine

$\frac{\sqrt{2} \cdot 9}{2}$ and

$\sqrt{2}$ .

$b = \frac{\sqrt{2} \cdot (9 \sqrt{2})}{2}$

Step 1.5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$b = \frac{9 (\sqrt{2} \sqrt{2})}{2}$

Step 1.5.4

Raise

$\sqrt{2}$ to the power of

$1$ .

$b = \frac{9 (\sqrt{2} \sqrt{2})}{2}$

Step 1.5.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b = \frac{9 {\sqrt{2}}^{1 + 1}}{2}$

Step 1.5.6

Add

$1$ and

$1$ .

$b = \frac{9 {\sqrt{2}}^{2}}{2}$

Step 1.6

Rewrite

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

$2$ .

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Step 1.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$b = \frac{9 {(2^{\frac{1}{2}})}^{2}}{2}$

Step 1.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$b = \frac{9 \cdot 2^{\frac{1}{2} \cdot 2}}{2}$

Step 1.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$b = \frac{9 \cdot 2^{\frac{2}{2}}}{2}$

Step 1.6.4

Cancel the common factor of

$2$ .

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Step 1.6.4.1

Cancel the common factor.

$b = \frac{9 \cdot 2^{\frac{2}{2}}}{2}$

Step 1.6.4.2

Rewrite the expression.

$b = \frac{9 \cdot 2}{2}$

Step 1.6.5

Evaluate the exponent.

$b = \frac{9 \cdot 2}{2}$

Step 1.7

Simplify the expression.

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Step 1.7.1

Multiply

$9$ by

$2$ .

$b = \frac{18}{2}$

Step 1.7.2

Divide

$18$ by

$2$ .

$b = 9$

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

$a$ .

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

Step 2.3

Substitute the actual values into the equation.

$a = \sqrt{{(9 \sqrt{2})}^{2} - {(9)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Apply the product rule to

$9 \sqrt{2}$ .

$a = \sqrt{9^{2} {\sqrt{2}}^{2} - {(9)}^{2}}$

Step 2.4.2

Raise

$9$ to the power of

$2$ .

$a = \sqrt{81 {\sqrt{2}}^{2} - {(9)}^{2}}$

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

$2^{\frac{1}{2}}$ .

$a = \sqrt{81 {(2^{\frac{1}{2}})}^{2} - {(9)}^{2}}$

Step 2.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$a = \sqrt{81 \cdot 2^{\frac{1}{2} \cdot 2} - {(9)}^{2}}$

Step 2.5.3

Combine

$\frac{1}{2}$ and

$2$ .

$a = \sqrt{81 \cdot 2^{\frac{2}{2}} - {(9)}^{2}}$

Step 2.5.4

Cancel the common factor of

$2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{81 \cdot 2^{\frac{2}{2}} - {(9)}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$a = \sqrt{81 \cdot 2 - {(9)}^{2}}$

Step 2.5.5

Evaluate the exponent.

$a = \sqrt{81 \cdot 2 - {(9)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply

$81$ by

$2$ .

$a = \sqrt{162 - {(9)}^{2}}$

Step 2.6.2

Raise

$9$ to the power of

$2$ .

$a = \sqrt{162 - 1 \cdot 81}$

Step 2.6.3

Multiply

$- 1$ by

$81$ .

$a = \sqrt{162 - 81}$

Step 2.6.4

Subtract

$81$ from

$162$ .

$a = \sqrt{81}$

Step 2.6.5

Rewrite

$81$ as

$9^{2}$ .

$a = \sqrt{9^{2}}$

Step 2.6.6

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

$a = 9$

Step 3

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

$A = 45$

$B = 45$

$C = 90$

$a = 9$

$b = 9$

$c = 9 \sqrt{2}$