Precalculus Examples

$\sin (2 x) = - \frac{4}{5}$

Step 1

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

$2 x = \arcsin (- \frac{4}{5})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (- \frac{4}{5})$ .

$2 x = - 0.92729521$

Step 3

Divide each term in $2 x = - 0.92729521$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = - 0.92729521$ by $2$ .

$\frac{2 x}{2} = \frac{- 0.92729521}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 0.92729521}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.92729521}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 0.92729521$ by $2$ .

$x = - 0.4636476$

Step 4

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.

$2 x = 2 (3.14159265) + 0.92729521 + 3.14159265$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract $2 π$ from $2 (3.14159265) + 0.92729521 + 3.14159265$ .

$2 x = 2 (3.14159265) + 0.92729521 + 3.14159265 - 2 π$

Step 5.2

The resulting angle of $4.06888787$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.92729521 + 3.14159265$ .

$2 x = 4.06888787$

Step 5.3

Divide each term in $2 x = 4.06888787$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 x = 4.06888787$ by $2$ .

$\frac{2 x}{2} = \frac{4.06888787}{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.06888787}{2}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.06888787}{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $4.06888787$ by $2$ .

$x = 2.03444393$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 2 |}$

Step 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 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide $π$ by $1$ .

$π$

Step 7

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

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Step 7.1

Add $π$ to $- 0.4636476$ to find the positive angle.

$- 0.4636476 + π$

Step 7.2

Replace with decimal approximation.

$3.14159265 - 0.4636476$

Step 7.3

Subtract $0.4636476$ from $3.14159265$ .

$2.67794504$

Step 7.4

List the new angles.

$x = 2.67794504$

Step 8

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

$x = 2.03444393 + π n, 2.67794504 + π n$ , for any integer $n$