Trigonometry Examples

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

Step 1

Divide each term in $2 \sin (2 x - 1) = 0$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \sin (2 x - 1) = 0$ by $2$ .

$\frac{2 \sin (2 x - 1)}{2} = \frac{0}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \sin (2 x - 1)}{2} = \frac{0}{2}$

Step 1.2.1.2

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

$\sin (2 x - 1) = \frac{0}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $0$ by $2$ .

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

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

$2 x - 1 = 0$

Step 4

Add $1$ to both sides of the equation.

$2 x = 1$

Step 5

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

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.

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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.

$2 x - 1 = π - 0$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $0$ from $π$ .

$2 x - 1 = π$

Step 7.2

Add $1$ to both sides of the equation.

$2 x = π + 1$

Step 7.3

Divide each term in $2 x = π + 1$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x = π + 1$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} + \frac{1}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} + \frac{1}{2}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2} + \frac{1}{2}$

Step 8

Find the period of $\sin (2 x - 1)$ .

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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 $\sin (2 x - 1)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{1}{2} + π n, \frac{π}{2} + \frac{1}{2} + π n$ , for any integer $n$

Step 10

Consolidate $\frac{1}{2} + π n$ and $\frac{π}{2} + \frac{1}{2} + π n$ to $\frac{1}{2} + \frac{π n}{2}$ .

$x = \frac{1}{2} + \frac{π n}{2}$ , for any integer $n$