Calculus Examples

$\sin (x^{2}) = 0$

Step 1

Substitute $u$ for $x^{2}$ .

$\sin (u) = 0$

Step 2

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

$u = \arcsin (0)$

Step 3

Simplify the right side.

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Step 3.1

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

$u = 0$

Step 4

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.

$u = π - 0$

Step 5

Subtract $0$ from $π$ .

$u = π$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

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

Step 8

Consolidate the answers.

$u = π n$ , for any integer $n$

Step 9

Substitute $x^{2}$ for $u$ and solve $x^{2} = π n$

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Step 9.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{π n}$

Step 9.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 9.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{π n}$

Step 9.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{π n}$

Step 9.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{π n}, - \sqrt{π n}$