Trigonometry Examples

$\cos (B) = \frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1

Simplify $\frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

$\cos (B) = \frac{25 + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1.1.2

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

$\cos (B) = \frac{25 + 64 - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1.1.3

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

$\cos (B) = \frac{25 + 64 - 1 \cdot 64}{2 \cdot 5 \cdot 8}$

Step 1.1.4

Multiply $- 1$ by $64$ .

$\cos (B) = \frac{25 + 64 - 64}{2 \cdot 5 \cdot 8}$

Step 1.1.5

Add $25$ and $64$ .

$\cos (B) = \frac{89 - 64}{2 \cdot 5 \cdot 8}$

Step 1.1.6

Subtract $64$ from $89$ .

$\cos (B) = \frac{25}{2 \cdot 5 \cdot 8}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply $2$ by $5$ .

$\cos (B) = \frac{25}{10 \cdot 8}$

Step 1.2.2

Multiply $10$ by $8$ .

$\cos (B) = \frac{25}{80}$

Step 1.3

Cancel the common factor of $25$ and $80$ .

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Step 1.3.1

Factor $5$ out of $25$ .

$\cos (B) = \frac{5 (5)}{80}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $5$ out of $80$ .

$\cos (B) = \frac{5 \cdot 5}{5 \cdot 16}$

Step 1.3.2.2

Cancel the common factor.

$\cos (B) = \frac{5 \cdot 5}{5 \cdot 16}$

Step 1.3.2.3

Rewrite the expression.

$\cos (B) = \frac{5}{16}$

Step 2

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

$B = \arccos (\frac{5}{16})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{5}{16})$ .

$B = 1.25297262$

Step 4

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

$B = 2 (3.14159265) - 1.25297262$

Step 5

Simplify $2 (3.14159265) - 1.25297262$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$B = 6.2831853 - 1.25297262$

Step 5.2

Subtract $1.25297262$ from $6.2831853$ .

$B = 5.03021268$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$B = 1.25297262 + 2 π n, 5.03021268 + 2 π n$ , for any integer $n$