Precalculus Examples

$17^{2} = {(22)}^{2} + {(25)}^{2} - 2 \cdot 22 \cdot 25 \cos (a)$

Step 1

Rewrite the equation as ${(22)}^{2} + {(25)}^{2} - 2 \cdot 22 \cdot (25 \cos (a)) = 17^{2}$ .

${(22)}^{2} + {(25)}^{2} - 2 \cdot 22 \cdot (25 \cos (a)) = 17^{2}$

Step 2

Simplify $22^{2} + 25^{2} - 2 \cdot 22 \cdot (25 \cos (a))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $22$ to the power of $2$ .

$484 + 25^{2} - 2 \cdot 22 \cdot (25 \cos (a)) = 17^{2}$

Step 2.1.2

Raise $25$ to the power of $2$ .

$484 + 625 - 2 \cdot 22 \cdot (25 \cos (a)) = 17^{2}$

Step 2.1.3

Multiply $- 2$ by $22$ .

$484 + 625 - 44 \cdot (25 \cos (a)) = 17^{2}$

Step 2.1.4

Multiply $25$ by $- 44$ .

$484 + 625 - 1100 \cos (a) = 17^{2}$

Step 2.2

Add $484$ and $625$ .

$1109 - 1100 \cos (a) = 17^{2}$

Step 3

Raise $17$ to the power of $2$ .

$1109 - 1100 \cos (a) = 289$

Step 4

Move all terms not containing $\cos (a)$ to the right side of the equation.

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Step 4.1

Subtract $1109$ from both sides of the equation.

$- 1100 \cos (a) = 289 - 1109$

Step 4.2

Subtract $1109$ from $289$ .

$- 1100 \cos (a) = - 820$

Step 5

Divide each term in $- 1100 \cos (a) = - 820$ by $- 1100$ and simplify.

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Step 5.1

Divide each term in $- 1100 \cos (a) = - 820$ by $- 1100$ .

$\frac{- 1100 \cos (a)}{- 1100} = \frac{- 820}{- 1100}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 1100$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 1100 \cos (a)}{- 1100} = \frac{- 820}{- 1100}$

Step 5.2.1.2

Divide $\cos (a)$ by $1$ .

$\cos (a) = \frac{- 820}{- 1100}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 820$ and $- 1100$ .

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Step 5.3.1.1

Factor $- 20$ out of $- 820$ .

$\cos (a) = \frac{- 20 (41)}{- 1100}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 20$ out of $- 1100$ .

$\cos (a) = \frac{- 20 \cdot 41}{- 20 \cdot 55}$

Step 5.3.1.2.2

Cancel the common factor.

$\cos (a) = \frac{- 20 \cdot 41}{- 20 \cdot 55}$

Step 5.3.1.2.3

Rewrite the expression.

$\cos (a) = \frac{41}{55}$

Step 6

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

$a = \arccos (\frac{41}{55})$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (\frac{41}{55})$ .

$a = 0.72957981$

Step 8

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.

$a = 2 (3.14159265) - 0.72957981$

Step 9

Simplify $2 (3.14159265) - 0.72957981$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$a = 6.2831853 - 0.72957981$

Step 9.2

Subtract $0.72957981$ from $6.2831853$ .

$a = 5.55360549$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 1 |}$

Step 10.3

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

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

$a = 0.72957981 + 2 π n, 5.55360549 + 2 π n$ , for any integer $n$