Calculus Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \sin (2 x)$ . Substitute $u$ for all occurrences of $\sin (2 x)$ .

$2 u^{2} - u = 0$

Step 1.2

Factor $u$ out of $2 u^{2} - u$ .

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Step 1.2.1

Factor $u$ out of $2 u^{2}$ .

$u (2 u) - u = 0$

Step 1.2.2

Factor $u$ out of $- u$ .

$u (2 u) + u \cdot - 1 = 0$

Step 1.2.3

Factor $u$ out of $u (2 u) + u \cdot - 1$ .

$u (2 u - 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sin (2 x)$ .

$\sin (2 x) (2 \sin (2 x) - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (2 x) = 0$

$2 \sin (2 x) - 1 = 0$

Step 3

Set $\sin (2 x)$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\sin (2 x)$ equal to $0$ .

$\sin (2 x) = 0$

Step 3.2

Solve $\sin (2 x) = 0$ for $x$ .

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Step 3.2.1

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

$2 x = \arcsin (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

$2 x = 0$

Step 3.2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3.2.4

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.

$2 x = π - 0$

Step 3.2.5

Solve for $x$ .

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Step 3.2.5.1

Simplify.

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Step 3.2.5.1.1

Multiply $- 1$ by $0$ .

$2 x = π + 0$

Step 3.2.5.1.2

Add $π$ and $0$ .

$2 x = π$

Step 3.2.5.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 3.2.5.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.2.5.2.2

Simplify the left side.

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Step 3.2.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 3.2.6

Find the period of $\sin (2 x)$ .

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Step 3.2.6.1

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

$\frac{2 π}{| b |}$

Step 3.2.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.6.4.2

Divide $π$ by $1$ .

$π$

Step 3.2.7

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

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

Step 4

Set $2 \sin (2 x) - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 \sin (2 x) - 1$ equal to $0$ .

$2 \sin (2 x) - 1 = 0$

Step 4.2

Solve $2 \sin (2 x) - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$2 \sin (2 x) = 1$

Step 4.2.2

Divide each term in $2 \sin (2 x) = 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 \sin (2 x) = 1$ by $2$ .

$\frac{2 \sin (2 x)}{2} = \frac{1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (2 x)}{2} = \frac{1}{2}$

Step 4.2.2.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{1}{2}$

Step 4.2.3

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

$2 x = \arcsin (\frac{1}{2})$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$2 x = \frac{π}{6}$

Step 4.2.5

Divide each term in $2 x = \frac{π}{6}$ by $2$ and simplify.

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Step 4.2.5.1

Divide each term in $2 x = \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 4.2.5.2

Simplify the left side.

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Step 4.2.5.2.1

Cancel the common factor of $2$ .

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Step 4.2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 4.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{2}$

Step 4.2.5.3

Simplify the right side.

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Step 4.2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 4.2.5.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{2}$ .

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Step 4.2.5.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{2}$ .

$x = \frac{π}{6 \cdot 2}$

Step 4.2.5.3.2.2

Multiply $6$ by $2$ .

$x = \frac{π}{12}$

Step 4.2.6

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.

$2 x = π - \frac{π}{6}$

Step 4.2.7

Solve for $x$ .

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Step 4.2.7.1

Simplify.

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Step 4.2.7.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 4.2.7.1.2

Combine $π$ and $\frac{6}{6}$ .

$2 x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 4.2.7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 6 - π}{6}$

Step 4.2.7.1.4

Subtract $π$ from $π \cdot 6$ .

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Step 4.2.7.1.4.1

Reorder $π$ and $6$ .

$2 x = \frac{6 \cdot π - π}{6}$

Step 4.2.7.1.4.2

Subtract $π$ from $6 \cdot π$ .

$2 x = \frac{5 \cdot π}{6}$

Step 4.2.7.2

Divide each term in $2 x = \frac{5 \cdot π}{6}$ by $2$ and simplify.

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Step 4.2.7.2.1

Divide each term in $2 x = \frac{5 \cdot π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 4.2.7.2.2

Simplify the left side.

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Step 4.2.7.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 4.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{6}}{2}$

Step 4.2.7.2.3

Simplify the right side.

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Step 4.2.7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 \cdot π}{6} \cdot \frac{1}{2}$

Step 4.2.7.2.3.2

Multiply $\frac{5 π}{6} \cdot \frac{1}{2}$ .

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Step 4.2.7.2.3.2.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{6 \cdot 2}$

Step 4.2.7.2.3.2.2

Multiply $6$ by $2$ .

$x = \frac{5 π}{12}$

Step 4.2.8

Find the period of $\sin (2 x)$ .

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Step 4.2.8.1

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

$\frac{2 π}{| b |}$

Step 4.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 4.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.2.8.4

Cancel the common factor of $2$ .

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Step 4.2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.2.8.4.2

Divide $π$ by $1$ .

$π$

Step 4.2.9

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

$x = \frac{π}{12} + π n, \frac{5 π}{12} + π n$ , for any integer $n$

Step 5

The final solution is all the values that make $\sin (2 x) (2 \sin (2 x) - 1) = 0$ true.

$x = π n, \frac{π}{2} + π n, \frac{π}{12} + π n, \frac{5 π}{12} + π n$ , for any integer $n$

Step 6

Consolidate $π n$ and $\frac{π}{2} + π n$ to $\frac{π n}{2}$ .

$x = \frac{π n}{2}, \frac{π}{12} + π n, \frac{5 π}{12} + π n$ , for any integer $n$