Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$c = \frac{5}{\sin (60)}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$c = 5 (\frac{2}{\sqrt{3}})$

Step 1.6

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$c = 5 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$c = 5 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 1.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$c = 5 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 1.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$c = 5 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 1.7.4

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

$c = 5 (\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 1.7.5

Add $1$ and $1$ .

$c = 5 (\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})$

Step 1.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$c = 5 (\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 1.7.6.2

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

$c = 5 (\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 1.7.6.3

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

$c = 5 (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

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.

$c = 5 (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 1.7.6.4.2

Rewrite the expression.

$c = 5 (\frac{2 \sqrt{3}}{3})$

Step 1.7.6.5

Evaluate the exponent.

$c = 5 (\frac{2 \sqrt{3}}{3})$

Step 1.8

Multiply $5 \frac{2 \sqrt{3}}{3}$ .

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Step 1.8.1

Combine $5$ and $\frac{2 \sqrt{3}}{3}$ .

$c = \frac{5 (2 \sqrt{3})}{3}$

Step 1.8.2

Multiply $2$ by $5$ .

$c = \frac{10 \sqrt{3}}{3}$

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.

$a = \sqrt{{(\frac{10 \sqrt{3}}{3})}^{2} - {(5)}^{2}}$

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.1

Apply the product rule to $\frac{10 \sqrt{3}}{3}$ .

$a = \sqrt{\frac{{(10 \sqrt{3})}^{2}}{3^{2}} - {(5)}^{2}}$

Step 2.4.2

Apply the product rule to $10 \sqrt{3}$ .

$a = \sqrt{\frac{10^{2} {\sqrt{3}}^{2}}{3^{2}} - {(5)}^{2}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Raise $10$ to the power of $2$ .

$a = \sqrt{\frac{100 {\sqrt{3}}^{2}}{3^{2}} - {(5)}^{2}}$

Step 2.5.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$a = \sqrt{\frac{100 {(3^{\frac{1}{2}})}^{2}}{3^{2}} - {(5)}^{2}}$

Step 2.5.2.2

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

$a = \sqrt{\frac{100 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}} - {(5)}^{2}}$

Step 2.5.2.3

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

$a = \sqrt{\frac{100 \cdot 3^{\frac{2}{2}}}{3^{2}} - {(5)}^{2}}$

Step 2.5.2.4

Cancel the common factor of $2$ .

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Step 2.5.2.4.1

Cancel the common factor.

$a = \sqrt{\frac{100 \cdot 3^{\frac{2}{2}}}{3^{2}} - {(5)}^{2}}$

Step 2.5.2.4.2

Rewrite the expression.

$a = \sqrt{\frac{100 \cdot 3}{3^{2}} - {(5)}^{2}}$

Step 2.5.2.5

Evaluate the exponent.

$a = \sqrt{\frac{100 \cdot 3}{3^{2}} - {(5)}^{2}}$

Step 2.6

Reduce the expression by cancelling the common factors.

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Step 2.6.1

Raise $3$ to the power of $2$ .

$a = \sqrt{\frac{100 \cdot 3}{9} - {(5)}^{2}}$

Step 2.6.2

Multiply $100$ by $3$ .

$a = \sqrt{\frac{300}{9} - {(5)}^{2}}$

Step 2.6.3

Cancel the common factor of $300$ and $9$ .

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Step 2.6.3.1

Factor $3$ out of $300$ .

$a = \sqrt{\frac{3 (100)}{9} - {(5)}^{2}}$

Step 2.6.3.2

Cancel the common factors.

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Step 2.6.3.2.1

Factor $3$ out of $9$ .

$a = \sqrt{\frac{3 \cdot 100}{3 \cdot 3} - {(5)}^{2}}$

Step 2.6.3.2.2

Cancel the common factor.

$a = \sqrt{\frac{3 \cdot 100}{3 \cdot 3} - {(5)}^{2}}$

Step 2.6.3.2.3

Rewrite the expression.

$a = \sqrt{\frac{100}{3} - {(5)}^{2}}$

Step 2.6.4

Simplify the expression.

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Step 2.6.4.1

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

$a = \sqrt{\frac{100}{3} - 1 \cdot 25}$

Step 2.6.4.2

Multiply $- 1$ by $25$ .

$a = \sqrt{\frac{100}{3} - 25}$

Step 2.7

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = \sqrt{\frac{100}{3} - 25 \cdot \frac{3}{3}}$

Step 2.8

Combine $- 25$ and $\frac{3}{3}$ .

$a = \sqrt{\frac{100}{3} + \frac{- 25 \cdot 3}{3}}$

Step 2.9

Combine the numerators over the common denominator.

$a = \sqrt{\frac{100 - 25 \cdot 3}{3}}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 25$ by $3$ .

$a = \sqrt{\frac{100 - 75}{3}}$

Step 2.10.2

Subtract $75$ from $100$ .

$a = \sqrt{\frac{25}{3}}$

Step 2.11

Rewrite $\sqrt{\frac{25}{3}}$ as $\frac{\sqrt{25}}{\sqrt{3}}$ .

$a = \frac{\sqrt{25}}{\sqrt{3}}$

Step 2.12

Simplify the numerator.

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Step 2.12.1

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

$a = \frac{\sqrt{5^{2}}}{\sqrt{3}}$

Step 2.12.2

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

$a = \frac{5}{\sqrt{3}}$

Step 2.13

Multiply $\frac{5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$a = \frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.14

Combine and simplify the denominator.

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Step 2.14.1

Multiply $\frac{5}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$a = \frac{5 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.14.2

Raise $\sqrt{3}$ to the power of $1$ .

$a = \frac{5 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.14.3

Raise $\sqrt{3}$ to the power of $1$ .

$a = \frac{5 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.14.4

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

$a = \frac{5 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.14.5

Add $1$ and $1$ .

$a = \frac{5 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.14.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$a = \frac{5 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.14.6.2

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

$a = \frac{5 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.14.6.3

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

$a = \frac{5 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.14.6.4

Cancel the common factor of $2$ .

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Step 2.14.6.4.1

Cancel the common factor.

$a = \frac{5 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.14.6.4.2

Rewrite the expression.

$a = \frac{5 \sqrt{3}}{3}$

Step 2.14.6.5

Evaluate the exponent.

$a = \frac{5 \sqrt{3}}{3}$

Step 3

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

$A = 30$

$B = 60$

$C = 90$

$a = \frac{5 \sqrt{3}}{3}$

$b = 5$

$c = \frac{10 \sqrt{3}}{3}$