Trigonometry Examples

${(9.9)}^{2} = {(8.4)}^{2} + {(8.7)}^{2} - 2 \cdot 8.4 \cdot 8.7 \cos (c)$

Step 1

Rewrite the equation as ${(8.4)}^{2} + {(8.7)}^{2} - 2 \cdot 8.4 \cdot (8.7 \cos (c)) = {9.9}^{2}$ .

${(8.4)}^{2} + {(8.7)}^{2} - 2 \cdot 8.4 \cdot (8.7 \cos (c)) = {9.9}^{2}$

Step 2

Simplify ${8.4}^{2} + {8.7}^{2} - 2 \cdot 8.4 \cdot (8.7 \cos (c))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $8.4$ to the power of $2$ .

$70.56 + {8.7}^{2} - 2 \cdot 8.4 \cdot (8.7 \cos (c)) = {9.9}^{2}$

Step 2.1.2

Raise $8.7$ to the power of $2$ .

$70.56 + 75.69 - 2 \cdot 8.4 \cdot (8.7 \cos (c)) = {9.9}^{2}$

Step 2.1.3

Multiply $- 2$ by $8.4$ .

$70.56 + 75.69 - 16.8 \cdot (8.7 \cos (c)) = {9.9}^{2}$

Step 2.1.4

Multiply $8.7$ by $- 16.8$ .

$70.56 + 75.69 - 146.16 \cos (c) = {9.9}^{2}$

Step 2.2

Add $70.56$ and $75.69$ .

$146.25 - 146.16 \cos (c) = {9.9}^{2}$

Step 3

Raise $9.9$ to the power of $2$ .

$146.25 - 146.16 \cos (c) = 98.01$

Step 4

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

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Step 4.1

Subtract $146.25$ from both sides of the equation.

$- 146.16 \cos (c) = 98.01 - 146.25$

Step 4.2

Subtract $146.25$ from $98.01$ .

$- 146.16 \cos (c) = - 48.24$

Step 5

Divide each term in $- 146.16 \cos (c) = - 48.24$ by $- 146.16$ and simplify.

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Step 5.1

Divide each term in $- 146.16 \cos (c) = - 48.24$ by $- 146.16$ .

$\frac{- 146.16 \cos (c)}{- 146.16} = \frac{- 48.24}{- 146.16}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 146.16$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 146.16 \cos (c)}{- 146.16} = \frac{- 48.24}{- 146.16}$

Step 5.2.1.2

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

$\cos (c) = \frac{- 48.24}{- 146.16}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 48.24$ by $- 146.16$ .

$\cos (c) = 0.33004926$

Step 6

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

$c = \arccos (0.33004926)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (0.33004926)$ .

$c = 1.23444056$

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.

$c = 2 (3.14159265) - 1.23444056$

Step 9

Simplify $2 (3.14159265) - 1.23444056$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$c = 6.2831853 - 1.23444056$

Step 9.2

Subtract $1.23444056$ from $6.2831853$ .

$c = 5.04874474$

Step 10

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

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

$c = 1.23444056 + 2 π n, 5.04874474 + 2 π n$ , for any integer $n$