Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 8 \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find the last angle of the triangle.

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Step 1.1

The sum of all the angles in a triangle is $180$ degrees.

$30 + 90 + B = 180$

Step 1.2

Solve the equation for $B$ .

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Step 1.2.1

Add $30$ and $90$ .

$120 + B = 180$

Step 1.2.2

Move all terms not containing $B$ to the right side of the equation.

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Step 1.2.2.1

Subtract $120$ from both sides of the equation.

$B = 180 - 120$

Step 1.2.2.2

Subtract $120$ from $180$ .

$B = 60$

Step 2

Find $b$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$b = 8 \cdot \cos (30)$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Factor $2$ out of $8$ .

$b = 2 (4) \cdot \frac{\sqrt{3}}{2}$

Step 2.5.2

Cancel the common factor.

$b = 2 \cdot 4 \cdot \frac{\sqrt{3}}{2}$

Step 2.5.3

Rewrite the expression.

$b = 4 \cdot \sqrt{3}$

$b = 4 \sqrt{3}$

Step 3

Find the last side of the triangle using the Pythagorean theorem.

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Step 3.1

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 3.2

Solve the equation for $a$ .

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

Step 3.3

Substitute the actual values into the equation.

$a = \sqrt{{(8)}^{2} - {(4 \sqrt{3})}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Raise $8$ to the power of $2$ .

$a = \sqrt{64 - {(4 \sqrt{3})}^{2}}$

Step 3.4.2

Apply the product rule to $4 \sqrt{3}$ .

$a = \sqrt{64 - (4^{2} {\sqrt{3}}^{2})}$

Step 3.4.3

Raise $4$ to the power of $2$ .

$a = \sqrt{64 - (16 {\sqrt{3}}^{2})}$

Step 3.5

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

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Step 3.5.1

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

$a = \sqrt{64 - (16 {(3^{\frac{1}{2}})}^{2})}$

Step 3.5.2

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

$a = \sqrt{64 - (16 \cdot 3^{\frac{1}{2} \cdot 2})}$

Step 3.5.3

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

$a = \sqrt{64 - (16 \cdot 3^{\frac{2}{2}})}$

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

$a = \sqrt{64 - (16 \cdot 3^{\frac{2}{2}})}$

Step 3.5.4.2

Rewrite the expression.

$a = \sqrt{64 - (16 \cdot 3)}$

Step 3.5.5

Evaluate the exponent.

$a = \sqrt{64 - (16 \cdot 3)}$

Step 3.6

Multiply $- (16 \cdot 3)$ .

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Step 3.6.1

Multiply $16$ by $3$ .

$a = \sqrt{64 - 1 \cdot 48}$

Step 3.6.2

Multiply $- 1$ by $48$ .

$a = \sqrt{64 - 48}$

Step 3.7

Subtract $48$ from $64$ .

$a = \sqrt{16}$

Step 3.8

Rewrite $16$ as $4^{2}$ .

$a = \sqrt{4^{2}}$

Step 3.9

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

$a = 4$

Step 4

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

$A = 30$

$B = 60$

$C = 90$

$a = 4$

$b = 4 \sqrt{3}$

$c = 8$