Precalculus Examples

$\cos (A) = \frac{36 + 64 - 25}{2 \cdot 6 \cdot 8}$

Step 1

Simplify $\frac{36 + 64 - 25}{2 \cdot 6 \cdot 8}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Add $36$ and $64$ .

$\cos (A) = \frac{100 - 25}{2 \cdot 6 \cdot 8}$

Step 1.1.2

Subtract $25$ from $100$ .

$\cos (A) = \frac{75}{2 \cdot 6 \cdot 8}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply $2$ by $6$ .

$\cos (A) = \frac{75}{12 \cdot 8}$

Step 1.2.2

Multiply $12$ by $8$ .

$\cos (A) = \frac{75}{96}$

Step 1.3

Cancel the common factor of $75$ and $96$ .

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Step 1.3.1

Factor $3$ out of $75$ .

$\cos (A) = \frac{3 (25)}{96}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $3$ out of $96$ .

$\cos (A) = \frac{3 \cdot 25}{3 \cdot 32}$

Step 1.3.2.2

Cancel the common factor.

$\cos (A) = \frac{3 \cdot 25}{3 \cdot 32}$

Step 1.3.2.3

Rewrite the expression.

$\cos (A) = \frac{25}{32}$

Step 2

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

$A = \arccos (\frac{25}{32})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{25}{32})$ .

$A = 0.6741305$

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.

$A = 2 (3.14159265) - 0.6741305$

Step 5

Simplify $2 (3.14159265) - 0.6741305$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$A = 6.2831853 - 0.6741305$

Step 5.2

Subtract $0.6741305$ from $6.2831853$ .

$A = 5.6090548$

Step 6

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

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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 $\cos (A)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$A = 0.6741305 + 2 π n, 5.6090548 + 2 π n$ , for any integer $n$