Precalculus Examples

$\ln (\sin (x)) = 0$

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\sin (x))} = e^{0}$

Step 2

Rewrite $\ln (\sin (x)) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \sin (x)$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\sin (x) = e^{0}$ .

$\sin (x) = e^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

$\sin (x) = 1$

Step 3.3

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

$x = \arcsin (1)$

Step 3.4

Simplify the right side.

Tap for more steps...

Step 3.4.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 3.5

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.

$x = π - \frac{π}{2}$

Step 3.6

Simplify $π - \frac{π}{2}$ .

Tap for more steps...

Step 3.6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 3.6.2

Combine fractions.

Tap for more steps...

Step 3.6.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 3.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 3.6.3

Simplify the numerator.

Tap for more steps...

Step 3.6.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 3.6.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 3.7

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

Tap for more steps...

Step 3.7.1

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

$\frac{2 π}{| b |}$

Step 3.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.7.3

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

$\frac{2 π}{1}$

Step 3.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.8

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

$x = \frac{π}{2} + 2 π n$ , for any integer $n$