Finite Math Examples

$\sin (2 x) = - (\frac{\sqrt{\sqrt{3}}}{2})$

Step 1

Rewrite $\sqrt{\sqrt{3}}$ as $\sqrt[4]{3}$ .

$\sin (2 x) = - \frac{\sqrt[4]{3}}{2}$

Step 2

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

$2 x = \arcsin (- \frac{\sqrt[4]{3}}{2})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (- \frac{\sqrt[4]{3}}{2})$ .

$2 x = - 0.71820883$

Step 4

Divide each term in $2 x = - 0.71820883$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - 0.71820883$ by $2$ .

$\frac{2 x}{2} = \frac{- 0.71820883}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 0.71820883}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.71820883}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 0.71820883$ by $2$ .

$x = - 0.35910441$

Step 5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$2 x = 2 (3.14159265) + 0.71820883 + 3.14159265$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $2 π$ from $2 (3.14159265) + 0.71820883 + 3.14159265$ .

$2 x = 2 (3.14159265) + 0.71820883 + 3.14159265 - 2 π$

Step 6.2

The resulting angle of $3.85980148$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.71820883 + 3.14159265$ .

$2 x = 3.85980148$

Step 6.3

Divide each term in $2 x = 3.85980148$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 x = 3.85980148$ by $2$ .

$\frac{2 x}{2} = \frac{3.85980148}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3.85980148}{2}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.85980148}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $3.85980148$ by $2$ .

$x = 1.92990074$

Step 7

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

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 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 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

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Step 8.1

Add $π$ to $- 0.35910441$ to find the positive angle.

$- 0.35910441 + π$

Step 8.2

Replace with decimal approximation.

$3.14159265 - 0.35910441$

Step 8.3

Subtract $0.35910441$ from $3.14159265$ .

$2.78248823$

Step 8.4

List the new angles.

$x = 2.78248823$

Step 9

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

$x = 1.92990074 + π n, 2.78248823 + π n$ , for any integer $n$