Finite Math Examples

$\ln (\sin (x))$

Step 1

Set the argument in $\ln (\sin (x))$ greater than $0$ to find where the expression is defined.

$\sin (x) > 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x > \arcsin (0)$

Step 2.2

Simplify the right side.

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Step 2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x > 0$

Step 2.3

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

$x = π - 0$

Step 2.4

Subtract $0$ from $π$ .

$x = π$

Step 2.5

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

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Step 2.5.1

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

$\frac{2 π}{| b |}$

Step 2.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.5.3

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

$\frac{2 π}{1}$

Step 2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.6

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

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

Step 2.7

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 2.8

Identify the leading coefficient.

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Step 2.8.1

The leading term in a polynomial is the term with the highest degree.

$\sin (x)$

Step 2.8.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 2.9

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $\sin (x)$ is always greater than $0$ .

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4