Trigonometry Examples

${22.24}^{2} = 11^{2} + 21^{2} - 2 \cdot 11 \cdot 21 \cos (c)$

Step 1

Rewrite the equation as $11^{2} + 21^{2} - 2 \cdot 11 \cdot (21 \cos (c)) = {22.24}^{2}$ .

$11^{2} + 21^{2} - 2 \cdot 11 \cdot (21 \cos (c)) = {22.24}^{2}$

Step 2

Simplify $11^{2} + 21^{2} - 2 \cdot 11 \cdot (21 \cos (c))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $11$ to the power of $2$ .

$121 + 21^{2} - 2 \cdot 11 \cdot (21 \cos (c)) = {22.24}^{2}$

Step 2.1.2

Raise $21$ to the power of $2$ .

$121 + 441 - 2 \cdot 11 \cdot (21 \cos (c)) = {22.24}^{2}$

Step 2.1.3

Multiply $- 2$ by $11$ .

$121 + 441 - 22 \cdot (21 \cos (c)) = {22.24}^{2}$

Step 2.1.4

Multiply $21$ by $- 22$ .

$121 + 441 - 462 \cos (c) = {22.24}^{2}$

Step 2.2

Add $121$ and $441$ .

$562 - 462 \cos (c) = {22.24}^{2}$

Step 3

Raise $22.24$ to the power of $2$ .

$562 - 462 \cos (c) = 494.6176$

Step 4

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

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Step 4.1

Subtract $562$ from both sides of the equation.

$- 462 \cos (c) = 494.6176 - 562$

Step 4.2

Subtract $562$ from $494.6176$ .

$- 462 \cos (c) = - 67.3824$

Step 5

Divide each term in $- 462 \cos (c) = - 67.3824$ by $- 462$ and simplify.

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Step 5.1

Divide each term in $- 462 \cos (c) = - 67.3824$ by $- 462$ .

$\frac{- 462 \cos (c)}{- 462} = \frac{- 67.3824}{- 462}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 462$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 462 \cos (c)}{- 462} = \frac{- 67.3824}{- 462}$

Step 5.2.1.2

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

$\cos (c) = \frac{- 67.3824}{- 462}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 67.3824$ by $- 462$ .

$\cos (c) = 0.14584935$

Step 6

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

$c = \arccos (0.14584935)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (0.14584935)$ .

$c = 1.42442487$

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.

$c = 2 (3.14159265) - 1.42442487$

Step 9

Simplify $2 (3.14159265) - 1.42442487$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$c = 6.2831853 - 1.42442487$

Step 9.2

Subtract $1.42442487$ from $6.2831853$ .

$c = 4.85876042$

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.

$c = 1.42442487 + 2 π n, 4.85876042 + 2 π n$ , for any integer $n$