Solve for x sin(arcsin(x))=1

$\sin (\arcsin (x)) = 1$

Step 1

Take the inverse sine of both sides of the equation to extract $\arcsin (x)$ from inside the sine.

$\arcsin (x) = \arcsin (1)$

Step 2

Simplify the right side.

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

$\arcsin (x) = \frac{π}{2}$

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.

$\arcsin (x) = π - \frac{π}{2}$

Step 4

Simplify $π - \frac{π}{2}$ .

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

$\arcsin (x) = π \cdot \frac{2}{2} - \frac{π}{2}$

Combine fractions.

Combine $π$ and $\frac{2}{2}$ .

$\arcsin (x) = \frac{π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$\arcsin (x) = \frac{π \cdot 2 - π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$\arcsin (x) = \frac{2 \cdot π - π}{2}$

Subtract $π$ from $2 π$ .

$\arcsin (x) = \frac{π}{2}$

Step 5

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

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

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

Step 7

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

$x = \sin (\frac{π}{2} + 2 π n)$

Solve for x sin(arcsin(x))=1

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