Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 2 \sqrt{3} \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = & 60 \\ C = & 90 \end{matrix} \end{matrix}$

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.

$b = 2 \sqrt{3} \cdot \cos (30)$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 1.5.2

Cancel the common factor.

$b = 2 \sqrt{3} \cdot \frac{\sqrt{3}}{2}$

Step 1.5.3

Rewrite the expression.

$b = \sqrt{3} \cdot \sqrt{3}$

Step 1.6

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

$b = {\sqrt{3}}^{1 + 1}$

Step 1.7

Add $1$ and $1$ .

$b = {\sqrt{3}}^{2}$

Step 1.8

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

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Step 1.8.1

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

$b = {(3^{\frac{1}{2}})}^{2}$

Step 1.8.2

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

$b = 3^{\frac{1}{2} \cdot 2}$

Step 1.8.3

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

$b = 3^{\frac{2}{2}}$

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

$b = 3^{\frac{2}{2}}$

Step 1.8.4.2

Rewrite the expression.

$b = 3$

Step 1.8.5

Evaluate the exponent.

$b = 3$

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.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Apply the product rule to $2 \sqrt{3}$ .

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

Step 2.4.2

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

$a = \sqrt{4 {\sqrt{3}}^{2} - {(3)}^{2}}$

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 $3^{\frac{1}{2}}$ .

$a = \sqrt{4 {(3^{\frac{1}{2}})}^{2} - {(3)}^{2}}$

Step 2.5.2

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

$a = \sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} - {(3)}^{2}}$

Step 2.5.3

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

$a = \sqrt{4 \cdot 3^{\frac{2}{2}} - {(3)}^{2}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$a = \sqrt{4 \cdot 3^{\frac{2}{2}} - {(3)}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$a = \sqrt{4 \cdot 3 - {(3)}^{2}}$

Step 2.5.5

Evaluate the exponent.

$a = \sqrt{4 \cdot 3 - {(3)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $4$ by $3$ .

$a = \sqrt{12 - {(3)}^{2}}$

Step 2.6.2

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

$a = \sqrt{12 - 1 \cdot 9}$

Step 2.6.3

Multiply $- 1$ by $9$ .

$a = \sqrt{12 - 9}$

Step 2.6.4

Subtract $9$ from $12$ .

$a = \sqrt{3}$

Step 3

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

$A = 30$

$B = 60$

$C = 90$

$a = \sqrt{3}$

$b = 3$

$c = 2 \sqrt{3}$