Trigonometry Examples

$\cos (B) = \frac{{(7)}^{2} + {(10.35513349)}^{2} - {(12)}^{2}}{2 (7) (10.35513349)}$

Step 1

Simplify $\frac{7^{2} + {10.35513349}^{2} - 12^{2}}{2 (7) \cdot 10.35513349}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Raise $7$ to the power of $2$ .

$\cos (B) = \frac{49 + {10.35513349}^{2} - 12^{2}}{2 (7) \cdot 10.35513349}$

Step 1.1.2

Raise $10.35513349$ to the power of $2$ .

$\cos (B) = \frac{49 + 107.22878959 - 12^{2}}{2 (7) \cdot 10.35513349}$

Step 1.1.3

Raise $12$ to the power of $2$ .

$\cos (B) = \frac{49 + 107.22878959 - 1 \cdot 144}{2 (7) \cdot 10.35513349}$

Step 1.1.4

Multiply $- 1$ by $144$ .

$\cos (B) = \frac{49 + 107.22878959 - 144}{2 (7) \cdot 10.35513349}$

Step 1.1.5

Add $49$ and $107.22878959$ .

$\cos (B) = \frac{156.22878959 - 144}{2 (7) \cdot 10.35513349}$

Step 1.1.6

Subtract $144$ from $156.22878959$ .

$\cos (B) = \frac{12.22878959}{2 (7) \cdot 10.35513349}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply $2$ by $7$ .

$\cos (B) = \frac{12.22878959}{14 \cdot 10.35513349}$

Step 1.2.2

Multiply $14$ by $10.35513349$ .

$\cos (B) = \frac{12.22878959}{144.97186886}$

Step 1.3

Divide $12.22878959$ by $144.97186886$ .

$\cos (B) = 0.08435284$

Step 2

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

$B = \arccos (0.08435284)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (0.08435284)$ .

$B = 1.48634312$

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.

$B = 2 (3.14159265) - 1.48634312$

Step 5

Simplify $2 (3.14159265) - 1.48634312$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$B = 6.2831853 - 1.48634312$

Step 5.2

Subtract $1.48634312$ from $6.2831853$ .

$B = 4.79684218$

Step 6

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

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

$B = 1.48634312 + 2 π n, 4.79684218 + 2 π n$ , for any integer $n$