Solve the Inequality for x sin(x)<1

$\sin (x) < 1$

Step 1

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

$x < \arcsin (1)$

Step 2

Simplify the right side.

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

$x < \frac{π}{2}$

Step 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.

$x = π - \frac{π}{2}$

Step 4

Simplify $π - \frac{π}{2}$ .

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

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

Combine fractions.

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 5

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Step 6

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 $n$

Step 7

Use each root to create test intervals.

$\frac{π}{2} < x < \frac{5 π}{2}$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $\frac{π}{2} < x < \frac{5 π}{2}$ to see if it makes the inequality true.

Choose a value on the interval $\frac{π}{2} < x < \frac{5 π}{2}$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$\sin (5) < 1$

The left side $- 0.95892427$ is less than the right side $1$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$\frac{π}{2} < x < \frac{5 π}{2}$ True

Step 9

The solution consists of all of the true intervals.

$\frac{π}{2} + 2 π n < x < \frac{5 π}{2} + 2 π n$ , for any integer $n$

Step 10

Solve the Inequality for x sin(x)<1

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