Calculus Examples

$0 = - \sin (x)$

Step 1

Rewrite the equation as $- \sin (x) = 0$ .

$- \sin (x) = 0$

Step 2

Divide each term in $- \sin (x) = 0$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \sin (x) = 0$ by $- 1$ .

$\frac{- \sin (x)}{- 1} = \frac{0}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} = \frac{0}{- 1}$

Step 2.2.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{0}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 1$ .

$\sin (x) = 0$

Step 3

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

$x = \arcsin (0)$

Step 4

Simplify the right side.

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Step 4.1

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

$x = 0$

Step 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.

$x = π - 0$

Step 6

Subtract $0$ from $π$ .

$x = π$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 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 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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 9

Consolidate the answers.

$x = π n$ , for any integer $n$