Trigonometry Examples

$\sin (2 x + \sqrt{3}) \sin (x) = 0$

Step 1

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

$\sin (2 x + \sqrt{3}) = 0$

$\sin (x) = 0$

Step 2

Set $\sin (2 x + \sqrt{3})$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\sin (2 x + \sqrt{3})$ equal to $0$ .

$\sin (2 x + \sqrt{3}) = 0$

Step 2.2

Solve $\sin (2 x + \sqrt{3}) = 0$ for $x$ .

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Step 2.2.1

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

$2 x + \sqrt{3} = \arcsin (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

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

$2 x + \sqrt{3} = 0$

Step 2.2.3

Subtract $\sqrt{3}$ from both sides of the equation.

$2 x = - \sqrt{3}$

Step 2.2.4

Divide each term in $2 x = - \sqrt{3}$ by $2$ and simplify.

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Step 2.2.4.1

Divide each term in $2 x = - \sqrt{3}$ by $2$ .

$\frac{2 x}{2} = \frac{- \sqrt{3}}{2}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \sqrt{3}}{2}$

Step 2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \sqrt{3}}{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{\sqrt{3}}{2}$

Step 2.2.5

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.

$2 x + \sqrt{3} = π - 0$

Step 2.2.6

Solve for $x$ .

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Step 2.2.6.1

Subtract $0$ from $π$ .

$2 x + \sqrt{3} = π$

Step 2.2.6.2

Subtract $\sqrt{3}$ from both sides of the equation.

$2 x = π - \sqrt{3}$

Step 2.2.6.3

Divide each term in $2 x = π - \sqrt{3}$ by $2$ and simplify.

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Step 2.2.6.3.1

Divide each term in $2 x = π - \sqrt{3}$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} + \frac{- \sqrt{3}}{2}$

Step 2.2.6.3.2

Simplify the left side.

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Step 2.2.6.3.2.1

Cancel the common factor of $2$ .

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Step 2.2.6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} + \frac{- \sqrt{3}}{2}$

Step 2.2.6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2} + \frac{- \sqrt{3}}{2}$

Step 2.2.6.3.3

Simplify the right side.

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Step 2.2.6.3.3.1

Move the negative in front of the fraction.

$x = \frac{π}{2} - \frac{\sqrt{3}}{2}$

Step 2.2.7

Find the period of $\sin (2 x + \sqrt{3})$ .

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Step 2.2.7.1

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

$\frac{2 π}{| b |}$

Step 2.2.7.2

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

$\frac{2 π}{| 2 |}$

Step 2.2.7.3

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

$\frac{2 π}{2}$

Step 2.2.7.4

Cancel the common factor of $2$ .

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Step 2.2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.2.7.4.2

Divide $π$ by $1$ .

$π$

Step 2.2.8

Add $π$ to every negative angle to get positive angles.

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Step 2.2.8.1

Add $π$ to $- \frac{\sqrt{3}}{2}$ to find the positive angle.

$- \frac{\sqrt{3}}{2} + π$

Step 2.2.8.2

List the new angles.

$x = - \frac{\sqrt{3}}{2} + π$

Step 2.2.9

The period of the $\sin (2 x + \sqrt{3})$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = - \frac{\sqrt{3}}{2} + π + π n, \frac{π}{2} - \frac{\sqrt{3}}{2} + π + π n$ , for any integer $n$

Step 3

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

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Step 3.1

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

$\sin (x) = 0$

Step 3.2

Solve $\sin (x) = 0$ 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.

$x = \arcsin (0)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

$x = 0$

Step 3.2.3

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 3.2.4

Subtract $0$ from $π$ .

$x = π$

Step 3.2.5

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

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Step 3.2.5.1

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

$\frac{2 π}{| b |}$

Step 3.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 3.2.5.3

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

$\frac{2 π}{1}$

Step 3.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.6

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 4

The final solution is all the values that make $\sin (2 x + \sqrt{3}) \sin (x) = 0$ true.

$x = - \frac{\sqrt{3}}{2} + π + π n, \frac{π}{2} - \frac{\sqrt{3}}{2} + π + π n, 2 π n, π + 2 π n$ , for any integer $n$

Step 5

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

$x = - \frac{\sqrt{3}}{2} + π + π n, \frac{π}{2} - \frac{\sqrt{3}}{2} + π + π n, π n$ , for any integer $n$