Trigonometry Examples

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

Step 1

Find

$c$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

$c$ .

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

Step 1.4

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

$c = \frac{2}{\sin (45)}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.6

Multiply

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

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

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

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply

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

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

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

Step 1.7.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 1.7.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 1.7.4

Use the power rule

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

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

Step 1.7.5

Add

$1$ and

$1$ .

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

Step 1.7.6

Rewrite

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

$2$ .

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Step 1.7.6.1

Use

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

$\sqrt{2}$ as

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

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

Step 1.7.6.2

Apply the power rule and multiply exponents,

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

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

Step 1.7.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 1.7.6.4

Cancel the common factor of

$2$ .

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Step 1.7.6.4.1

Cancel the common factor.

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

Step 1.7.6.4.2

Rewrite the expression.

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

Step 1.7.6.5

Evaluate the exponent.

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

Step 1.8

Cancel the common factor of

$2$ .

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Step 1.8.1

Cancel the common factor.

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

Step 1.8.2

Rewrite the expression.

$c = 2 \sqrt{2}$

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

$a$ .

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

Step 2.3

Substitute the actual values into the equation.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Apply the product rule to

$2 \sqrt{2}$ .

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

Step 2.4.2

Raise

$2$ to the power of

$2$ .

$a = \sqrt{4 {\sqrt{2}}^{2} - {(2)}^{2}}$

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

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

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

Step 2.5.2

Apply the power rule and multiply exponents,

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

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

Step 2.5.3

Combine

$\frac{1}{2}$ and

$2$ .

$a = \sqrt{4 \cdot 2^{\frac{2}{2}} - {(2)}^{2}}$

Step 2.5.4

Cancel the common factor of

$2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{4 \cdot 2^{\frac{2}{2}} - {(2)}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$a = \sqrt{4 \cdot 2 - {(2)}^{2}}$

Step 2.5.5

Evaluate the exponent.

$a = \sqrt{4 \cdot 2 - {(2)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply

$4$ by

$2$ .

$a = \sqrt{8 - {(2)}^{2}}$

Step 2.6.2

Raise

$2$ to the power of

$2$ .

$a = \sqrt{8 - 1 \cdot 4}$

Step 2.6.3

Multiply

$- 1$ by

$4$ .

$a = \sqrt{8 - 4}$

Step 2.6.4

Subtract

$4$ from

$8$ .

$a = \sqrt{4}$

Step 2.6.5

Rewrite

$4$ as

$2^{2}$ .

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

Step 2.7

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

$a = 2$

Step 3

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

$A = 45$

$B = 45$

$C = 90$

$a = 2$

$b = 2$

$c = 2 \sqrt{2}$