Precalculus Examples

$55^{2} = 50^{2} + 35^{2} - 2 \cdot 50 \cdot 35 \cos (A)$

Step 1

Rewrite the equation as $50^{2} + 35^{2} - 2 \cdot 50 \cdot (35 \cos (A)) = 55^{2}$ .

$50^{2} + 35^{2} - 2 \cdot 50 \cdot (35 \cos (A)) = 55^{2}$

Step 2

Simplify $50^{2} + 35^{2} - 2 \cdot 50 \cdot (35 \cos (A))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $50$ to the power of $2$ .

$2500 + 35^{2} - 2 \cdot 50 \cdot (35 \cos (A)) = 55^{2}$

Step 2.1.2

Raise $35$ to the power of $2$ .

$2500 + 1225 - 2 \cdot 50 \cdot (35 \cos (A)) = 55^{2}$

Step 2.1.3

Multiply $- 2$ by $50$ .

$2500 + 1225 - 100 \cdot (35 \cos (A)) = 55^{2}$

Step 2.1.4

Multiply $35$ by $- 100$ .

$2500 + 1225 - 3500 \cos (A) = 55^{2}$

Step 2.2

Add $2500$ and $1225$ .

$3725 - 3500 \cos (A) = 55^{2}$

Step 3

Raise $55$ to the power of $2$ .

$3725 - 3500 \cos (A) = 3025$

Step 4

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

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Step 4.1

Subtract $3725$ from both sides of the equation.

$- 3500 \cos (A) = 3025 - 3725$

Step 4.2

Subtract $3725$ from $3025$ .

$- 3500 \cos (A) = - 700$

Step 5

Divide each term in $- 3500 \cos (A) = - 700$ by $- 3500$ and simplify.

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Step 5.1

Divide each term in $- 3500 \cos (A) = - 700$ by $- 3500$ .

$\frac{- 3500 \cos (A)}{- 3500} = \frac{- 700}{- 3500}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 3500$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 3500 \cos (A)}{- 3500} = \frac{- 700}{- 3500}$

Step 5.2.1.2

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

$\cos (A) = \frac{- 700}{- 3500}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 700$ and $- 3500$ .

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Step 5.3.1.1

Factor $- 700$ out of $- 700$ .

$\cos (A) = \frac{- 700 (1)}{- 3500}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 700$ out of $- 3500$ .

$\cos (A) = \frac{- 700 \cdot 1}{- 700 \cdot 5}$

Step 5.3.1.2.2

Cancel the common factor.

$\cos (A) = \frac{- 700 \cdot 1}{- 700 \cdot 5}$

Step 5.3.1.2.3

Rewrite the expression.

$\cos (A) = \frac{1}{5}$

Step 6

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

$A = \arccos (\frac{1}{5})$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (\frac{1}{5})$ .

$A = 1.3694384$

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.

$A = 2 (3.14159265) - 1.3694384$

Step 9

Simplify $2 (3.14159265) - 1.3694384$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$A = 6.2831853 - 1.3694384$

Step 9.2

Subtract $1.3694384$ from $6.2831853$ .

$A = 4.9137469$

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.

$A = 1.3694384 + 2 π n, 4.9137469 + 2 π n$ , for any integer $n$