Trigonometry Examples

$2 \sin (2 x) - 1 = 0$ , $[0, 2 π)$

Step 1

Add $1$ to both sides of the equation.

$2 \sin (2 x) = 1$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 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

Simplify the right side.

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Step 4.1

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

$2 x = \frac{π}{6}$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 5.3.2

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

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Step 5.3.2.1

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

$x = \frac{π}{6 \cdot 2}$

Step 5.3.2.2

Multiply $6$ by $2$ .

$x = \frac{π}{12}$

Step 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 7

Solve for $x$ .

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Step 7.1

Simplify.

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Step 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 7.1.2

Combine $π$ and $\frac{6}{6}$ .

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

Step 7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 6 - π}{6}$

Step 7.1.4

Subtract $π$ from $π \cdot 6$ .

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Step 7.1.4.1

Reorder $π$ and $6$ .

$2 x = \frac{6 \cdot π - π}{6}$

Step 7.1.4.2

Subtract $π$ from $6 \cdot π$ .

$2 x = \frac{5 \cdot π}{6}$

Step 7.2

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

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Step 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 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 \cdot π}{6}}{2}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.3.2

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

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Step 7.2.3.2.1

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

$x = \frac{5 π}{6 \cdot 2}$

Step 7.2.3.2.2

Multiply $6$ by $2$ .

$x = \frac{5 π}{12}$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 2 |}$

Step 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 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 8.4.2

Divide $π$ by $1$ .

$π$

Step 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 10

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 10.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 10.1.1

Plug in $0$ for $n$ .

$\frac{π}{12} + π (0)$

Step 10.1.2

Simplify.

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Step 10.1.2.1

Multiply $π$ by $0$ .

$\frac{π}{12} + 0$

Step 10.1.2.2

Add $\frac{π}{12}$ and $0$ .

$\frac{π}{12}$

Step 10.1.3

The interval $[0, 2 π)$ contains $\frac{π}{12}$ .

$x = \frac{π}{12}$

Step 10.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 10.2.1

Plug in $0$ for $n$ .

$\frac{5 π}{12} + π (0)$

Step 10.2.2

Simplify.

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Step 10.2.2.1

Multiply $π$ by $0$ .

$\frac{5 π}{12} + 0$

Step 10.2.2.2

Add $\frac{5 π}{12}$ and $0$ .

$\frac{5 π}{12}$

Step 10.2.3

The interval $[0, 2 π)$ contains $\frac{5 π}{12}$ .

$x = \frac{π}{12}, \frac{5 π}{12}$

Step 10.3

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 10.3.1

Plug in $1$ for $n$ .

$\frac{π}{12} + π (1)$

Step 10.3.2

Simplify.

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Step 10.3.2.1

Multiply $π$ by $1$ .

$\frac{π}{12} + π$

Step 10.3.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{π}{12} + π \cdot \frac{12}{12}$

Step 10.3.2.3

Combine fractions.

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Step 10.3.2.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π}{12} + \frac{π \cdot 12}{12}$

Step 10.3.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 12}{12}$

Step 10.3.2.4

Simplify the numerator.

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Step 10.3.2.4.1

Move $12$ to the left of $π$ .

$\frac{π + 12 \cdot π}{12}$

Step 10.3.2.4.2

Add $π$ and $12 π$ .

$\frac{13 π}{12}$

Step 10.3.3

The interval $[0, 2 π)$ contains $\frac{13 π}{12}$ .

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}$

Step 10.4

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 10.4.1

Plug in $1$ for $n$ .

$\frac{5 π}{12} + π (1)$

Step 10.4.2

Simplify.

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Step 10.4.2.1

Multiply $π$ by $1$ .

$\frac{5 π}{12} + π$

Step 10.4.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5 π}{12} + π \cdot \frac{12}{12}$

Step 10.4.2.3

Combine fractions.

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Step 10.4.2.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{5 π}{12} + \frac{π \cdot 12}{12}$

Step 10.4.2.3.2

Combine the numerators over the common denominator.

$\frac{5 π + π \cdot 12}{12}$

Step 10.4.2.4

Simplify the numerator.

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Step 10.4.2.4.1

Move $12$ to the left of $π$ .

$\frac{5 π + 12 \cdot π}{12}$

Step 10.4.2.4.2

Add $5 π$ and $12 π$ .

$\frac{17 π}{12}$

Step 10.4.3

The interval $[0, 2 π)$ contains $\frac{17 π}{12}$ .

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$