Trigonometry Examples

cos (B) = \frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1

Simplify

\frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}

$\frac{5^{2} + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Raise

5

$5$ to the power of

2

$2$ .

cos (B) = \frac{25 + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25 + 8^{2} - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1.1.2

Raise

8

$8$ to the power of

2

$2$ .

cos (B) = \frac{25 + 64 - 8^{2}}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25 + 64 - 8^{2}}{2 \cdot 5 \cdot 8}$

Step 1.1.3

Raise

8

$8$ to the power of

2

$2$ .

cos (B) = \frac{25 + 64 - 1 \cdot 64}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25 + 64 - 1 \cdot 64}{2 \cdot 5 \cdot 8}$

Step 1.1.4

Multiply

- 1

$- 1$ by

64

$64$ .

cos (B) = \frac{25 + 64 - 64}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25 + 64 - 64}{2 \cdot 5 \cdot 8}$

Step 1.1.5

Add

25

$25$ and

64

$64$ .

cos (B) = \frac{89 - 64}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{89 - 64}{2 \cdot 5 \cdot 8}$

Step 1.1.6

Subtract

64

$64$ from

89

$89$ .

cos (B) = \frac{25}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25}{2 \cdot 5 \cdot 8}$

cos (B) = \frac{25}{2 \cdot 5 \cdot 8}

$\cos (B) = \frac{25}{2 \cdot 5 \cdot 8}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Multiply

2

$2$ by

5

$5$ .

cos (B) = \frac{25}{10 \cdot 8}

$\cos (B) = \frac{25}{10 \cdot 8}$

Step 1.2.2

Multiply

10

$10$ by

8

$8$ .

cos (B) = \frac{25}{80}

$\cos (B) = \frac{25}{80}$

cos (B) = \frac{25}{80}

$\cos (B) = \frac{25}{80}$

Step 1.3

Cancel the common factor of

25

$25$ and

80

$80$ .

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Step 1.3.1

Factor

5

$5$ out of

25

$25$ .

cos (B) = \frac{5 (5)}{80}

$\cos (B) = \frac{5 (5)}{80}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor

5

$5$ out of

80

$80$ .

cos (B) = \frac{5 \cdot 5}{5 \cdot 16}

$\cos (B) = \frac{5 \cdot 5}{5 \cdot 16}$

Step 1.3.2.2

Cancel the common factor.

$\cos (B) = \frac{5 \cdot 5}{5 \cdot 16}$

Step 1.3.2.3

Rewrite the expression.

$\cos (B) = \frac{5}{16}$

Step 2

Take the inverse cosine of both sides of the equation to extract

$B$ from inside the cosine.

$B = \arccos (\frac{5}{16})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate

$\arccos (\frac{5}{16})$ .

$B = 1.25297262$

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.

$B = 2 (3.14159265) - 1.25297262$

Step 5

Simplify

$2 (3.14159265) - 1.25297262$ .

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Step 5.1

Multiply

$2$ by

$3.14159265$ .

$B = 6.2831853 - 1.25297262$

Step 5.2

Subtract

$1.25297262$ from

$6.2831853$ .

$B = 5.03021268$

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.

$B = 1.25297262 + 2 π n, 5.03021268 + 2 π n$ , for any integer

$n$