Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \sqrt{3} \\ c = \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$c = \frac{5 \sqrt{3}}{\sin (60)}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.6

Cancel the common factor of $\sqrt{3}$ .

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Step 1.6.1

Factor $\sqrt{3}$ out of $5 \sqrt{3}$ .

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

Step 1.6.2

Cancel the common factor.

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

Step 1.6.3

Rewrite the expression.

$c = 5 \cdot 2$

Step 1.7

Multiply $5$ by $2$ .

$c = 10$

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.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

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

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

Step 2.4.2

Apply the product rule to $5 \sqrt{3}$ .

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

Step 2.4.3

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

$a = \sqrt{100 - (25 {\sqrt{3}}^{2})}$

Step 2.5

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

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Step 2.5.1

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

$a = \sqrt{100 - (25 {(3^{\frac{1}{2}})}^{2})}$

Step 2.5.2

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

$a = \sqrt{100 - (25 \cdot 3^{\frac{1}{2} \cdot 2})}$

Step 2.5.3

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

$a = \sqrt{100 - (25 \cdot 3^{\frac{2}{2}})}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{100 - (25 \cdot 3^{\frac{2}{2}})}$

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Evaluate the exponent.

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

Step 2.6

Multiply $- (25 \cdot 3)$ .

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Step 2.6.1

Multiply $25$ by $3$ .

$a = \sqrt{100 - 1 \cdot 75}$

Step 2.6.2

Multiply $- 1$ by $75$ .

$a = \sqrt{100 - 75}$

Step 2.7

Subtract $75$ from $100$ .

$a = \sqrt{25}$

Step 2.8

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

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

Step 2.9

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

$a = 5$

Step 3

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

$A = 30$

$B = 60$

$C = 90$

$a = 5$

$b = 5 \sqrt{3}$

$c = 10$