Trigonometry Examples

$\frac{27^{2} - 12^{2} - 20^{2}}{- 2 (12) (20)} = \cos (B)$

Step 1

Rewrite the equation as $\cos (B) = \frac{27^{2} - 12^{2} - 20^{2}}{- 2 \cdot 12 \cdot 20}$ .

$\cos (B) = \frac{27^{2} - 12^{2} - 20^{2}}{- 2 \cdot 12 \cdot 20}$

Step 2

Simplify $\frac{27^{2} - 12^{2} - 20^{2}}{- 2 \cdot 12 \cdot 20}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Raise $27$ to the power of $2$ .

$\cos (B) = \frac{729 - 12^{2} - 20^{2}}{- 2 \cdot 12 \cdot 20}$

Step 2.1.2

Raise $12$ to the power of $2$ .

$\cos (B) = \frac{729 - 1 \cdot 144 - 20^{2}}{- 2 \cdot 12 \cdot 20}$

Step 2.1.3

Multiply $- 1$ by $144$ .

$\cos (B) = \frac{729 - 144 - 20^{2}}{- 2 \cdot 12 \cdot 20}$

Step 2.1.4

Raise $20$ to the power of $2$ .

$\cos (B) = \frac{729 - 144 - 1 \cdot 400}{- 2 \cdot 12 \cdot 20}$

Step 2.1.5

Multiply $- 1$ by $400$ .

$\cos (B) = \frac{729 - 144 - 400}{- 2 \cdot 12 \cdot 20}$

Step 2.1.6

Subtract $144$ from $729$ .

$\cos (B) = \frac{585 - 400}{- 2 \cdot 12 \cdot 20}$

Step 2.1.7

Subtract $400$ from $585$ .

$\cos (B) = \frac{185}{- 2 \cdot 12 \cdot 20}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 2$ by $12$ .

$\cos (B) = \frac{185}{- 24 \cdot 20}$

Step 2.2.2

Multiply $- 24$ by $20$ .

$\cos (B) = \frac{185}{- 480}$

Step 2.3

Reduce the expression by cancelling the common factors.

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Step 2.3.1

Cancel the common factor of $185$ and $- 480$ .

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Step 2.3.1.1

Factor $5$ out of $185$ .

$\cos (B) = \frac{5 (37)}{- 480}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $- 480$ .

$\cos (B) = \frac{5 \cdot 37}{5 \cdot - 96}$

Step 2.3.1.2.2

Cancel the common factor.

$\cos (B) = \frac{5 \cdot 37}{5 \cdot - 96}$

Step 2.3.1.2.3

Rewrite the expression.

$\cos (B) = \frac{37}{- 96}$

Step 2.3.2

Move the negative in front of the fraction.

$\cos (B) = - \frac{37}{96}$

Step 3

Take the inverse cosine of both sides of the equation to extract $B$ from inside the cosine.

$B = \arccos (- \frac{37}{96})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (- \frac{37}{96})$ .

$B = 1.96645565$

Step 5

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$B = 2 (3.14159265) - 1.96645565$

Step 6

Simplify $2 (3.14159265) - 1.96645565$ .

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Step 6.1

Multiply $2$ by $3.14159265$ .

$B = 6.2831853 - 1.96645565$

Step 6.2

Subtract $1.96645565$ from $6.2831853$ .

$B = 4.31672964$

Step 7

Find the period of $\cos (B)$ .

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Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

The period of the $\cos (B)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$B = 1.96645565 + 2 π n, 4.31672964 + 2 π n$ , for any integer $n$