Trigonometry Examples

\cos (x) = \frac{12}{13}

$\cos (x) = \frac{12}{13}$ ,

\sin (\frac{x}{2})

$\sin (\frac{x}{2})$

Step 1

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

x

$x$ from inside the cosine.

x = \arccos (\frac{12}{13})

$x = \arccos (\frac{12}{13})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate

\arccos (\frac{12}{13})

$\arccos (\frac{12}{13})$ .

x = 0.39479111

$x = 0.39479111$

x = 0.39479111

$x = 0.39479111$

Step 3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

2 π

$2 π$ to find the solution in the fourth quadrant.

x = 2 (3.14159265) - 0.39479111

$x = 2 (3.14159265) - 0.39479111$

Step 4

Solve for

x

$x$ .

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Step 4.1

Remove parentheses.

x = 2 (3.14159265) - 0.39479111

$x = 2 (3.14159265) - 0.39479111$

Step 4.2

Simplify

2 (3.14159265) - 0.39479111

$2 (3.14159265) - 0.39479111$ .

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Step 4.2.1

Multiply

2

$2$ by

3.14159265

$3.14159265$ .

x = 6.2831853 - 0.39479111

$x = 6.2831853 - 0.39479111$

Step 4.2.2

Subtract

0.39479111

$0.39479111$ from

6.2831853

$6.2831853$ .

x = 5.88839418

$x = 5.88839418$

x = 5.88839418

$x = 5.88839418$

x = 5.88839418

$x = 5.88839418$

Step 5

Find the period of

\cos (x)

$\cos (x)$ .

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Step 5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 6

The period of the

\cos (x)

$\cos (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = 0.39479111 + 2 π n, 5.88839418 + 2 π n

$x = 0.39479111 + 2 π n, 5.88839418 + 2 π n$ , for any integer

n

$n$

Step 7

Take the base solution.

x = 0.39479111

$x = 0.39479111$

Step 8

Replace the variable

x

$x$ with

0.39479111

$0.39479111$ in the expression.

\sin (\frac{0.39479111}{2})

$\sin (\frac{0.39479111}{2})$

Step 9

Divide

0.39479111

$0.39479111$ by

2

$2$ .

\sin (0.19739555)

$\sin (0.19739555)$

Step 10

The result can be shown in multiple forms.

Exact Form:

\sin (0.19739555)

$\sin (0.19739555)$

Decimal Form:

0.19611613 \dots

$0.19611613 \dots$