Trigonometry Examples

$\cos (x) = \frac{12}{13}$ , $\sin (\frac{x}{2})$

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

$x = 0.39479111$

Step 3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 0.39479111$

Step 4

Solve for $x$ .

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Step 4.1

Remove parentheses.

$x = 2 (3.14159265) - 0.39479111$

Step 4.2

Simplify $2 (3.14159265) - 0.39479111$ .

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Step 4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.39479111$

Step 4.2.2

Subtract $0.39479111$ from $6.2831853$ .

$x = 5.88839418$

Step 5

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

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Step 5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Step 7

Take the base solution.

$x = 0.39479111$

Step 8

Replace the variable $x$ with $0.39479111$ in the expression.

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

Step 9

Divide $0.39479111$ by $2$ .

$\sin (0.19739555)$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\sin (0.19739555)$

Decimal Form:

$0.19611613 \dots$