Trigonometry Examples

$\cos (c) = \frac{22^{2} + 19^{2} - 19^{2}}{2 (22) (19)}$

Step 1

Simplify $\frac{22^{2} + 19^{2} - 19^{2}}{2 (22) \cdot 19}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

$\cos (c) = \frac{484 + 19^{2} - 19^{2}}{2 (22) \cdot 19}$

Step 1.1.2

Raise $19$ to the power of $2$ .

$\cos (c) = \frac{484 + 361 - 19^{2}}{2 (22) \cdot 19}$

Step 1.1.3

Raise $19$ to the power of $2$ .

$\cos (c) = \frac{484 + 361 - 1 \cdot 361}{2 (22) \cdot 19}$

Step 1.1.4

Multiply $- 1$ by $361$ .

$\cos (c) = \frac{484 + 361 - 361}{2 (22) \cdot 19}$

Step 1.1.5

Add $484$ and $361$ .

$\cos (c) = \frac{845 - 361}{2 (22) \cdot 19}$

Step 1.1.6

Subtract $361$ from $845$ .

$\cos (c) = \frac{484}{2 (22) \cdot 19}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply $2$ by $22$ .

$\cos (c) = \frac{484}{44 \cdot 19}$

Step 1.2.2

Multiply $44$ by $19$ .

$\cos (c) = \frac{484}{836}$

Step 1.3

Cancel the common factor of $484$ and $836$ .

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Step 1.3.1

Factor $44$ out of $484$ .

$\cos (c) = \frac{44 (11)}{836}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $44$ out of $836$ .

$\cos (c) = \frac{44 \cdot 11}{44 \cdot 19}$

Step 1.3.2.2

Cancel the common factor.

$\cos (c) = \frac{44 \cdot 11}{44 \cdot 19}$

Step 1.3.2.3

Rewrite the expression.

$\cos (c) = \frac{11}{19}$

Step 2

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

$c = \arccos (\frac{11}{19})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{11}{19})$ .

$c = 0.95335922$

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.

$c = 2 (3.14159265) - 0.95335922$

Step 5

Simplify $2 (3.14159265) - 0.95335922$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$c = 6.2831853 - 0.95335922$

Step 5.2

Subtract $0.95335922$ from $6.2831853$ .

$c = 5.32982608$

Step 6

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

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

$c = 0.95335922 + 2 π n, 5.32982608 + 2 π n$ , for any integer $n$