Precalculus Examples

$2 \sin (x - \frac{π}{4}) + \sqrt{3} = 0$

Step 1

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

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

Step 2

Divide each term in $2 \sin (x - \frac{π}{4}) = - \sqrt{3}$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \sin (x - \frac{π}{4}) = - \sqrt{3}$ by $2$ .

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

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.

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

Step 2.2.1.2

Divide $\sin (x - \frac{π}{4})$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arcsin (- \frac{\sqrt{3}}{2})$ is $- \frac{π}{3}$ .

$x - \frac{π}{4} = - \frac{π}{3}$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Add $\frac{π}{4}$ to both sides of the equation.

$x = - \frac{π}{3} + \frac{π}{4}$

Step 5.2

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - \frac{π}{3} \cdot \frac{4}{4} + \frac{π}{4}$

Step 5.3

To write $\frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{π}{3} \cdot \frac{4}{4} + \frac{π}{4} \cdot \frac{3}{3}$

Step 5.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{π}{3}$ by $\frac{4}{4}$ .

$x = - \frac{π \cdot 4}{3 \cdot 4} + \frac{π}{4} \cdot \frac{3}{3}$

Step 5.4.2

Multiply $3$ by $4$ .

$x = - \frac{π \cdot 4}{12} + \frac{π}{4} \cdot \frac{3}{3}$

Step 5.4.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$x = - \frac{π \cdot 4}{12} + \frac{π \cdot 3}{4 \cdot 3}$

Step 5.4.4

Multiply $4$ by $3$ .

$x = - \frac{π \cdot 4}{12} + \frac{π \cdot 3}{12}$

Step 5.5

Combine the numerators over the common denominator.

$x = \frac{- π \cdot 4 + π \cdot 3}{12}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $4$ by $- 1$ .

$x = \frac{- 4 π + π \cdot 3}{12}$

Step 5.6.2

Move $3$ to the left of $π$ .

$x = \frac{- 4 π + 3 \cdot π}{12}$

Step 5.6.3

Add $- 4 π$ and $3 π$ .

$x = \frac{- π}{12}$

Step 5.7

Move the negative in front of the fraction.

$x = - \frac{π}{12}$

Step 6

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.

$x - \frac{π}{4} = 2 π + \frac{π}{3} + π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $2 π$ from $2 π + \frac{π}{3} + π$ .

$x - \frac{π}{4} = 2 π + \frac{π}{3} + π - 2 π$

Step 7.2

The resulting angle of $\frac{4 π}{3}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{3} + π$ .

$x - \frac{π}{4} = \frac{4 π}{3}$

Step 7.3

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.1

Add $\frac{π}{4}$ to both sides of the equation.

$x = \frac{4 π}{3} + \frac{π}{4}$

Step 7.3.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{4 π}{3} \cdot \frac{4}{4} + \frac{π}{4}$

Step 7.3.3

To write $\frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{4 π}{3} \cdot \frac{4}{4} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.4.1

Multiply $\frac{4 π}{3}$ by $\frac{4}{4}$ .

$x = \frac{4 π \cdot 4}{3 \cdot 4} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.3.4.2

Multiply $3$ by $4$ .

$x = \frac{4 π \cdot 4}{12} + \frac{π}{4} \cdot \frac{3}{3}$

Step 7.3.4.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$x = \frac{4 π \cdot 4}{12} + \frac{π \cdot 3}{4 \cdot 3}$

Step 7.3.4.4

Multiply $4$ by $3$ .

$x = \frac{4 π \cdot 4}{12} + \frac{π \cdot 3}{12}$

Step 7.3.5

Combine the numerators over the common denominator.

$x = \frac{4 π \cdot 4 + π \cdot 3}{12}$

Step 7.3.6

Simplify the numerator.

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Step 7.3.6.1

Multiply $4$ by $4$ .

$x = \frac{16 π + π \cdot 3}{12}$

Step 7.3.6.2

Move $3$ to the left of $π$ .

$x = \frac{16 π + 3 \cdot π}{12}$

Step 7.3.6.3

Add $16 π$ and $3 π$ .

$x = \frac{19 π}{12}$

Step 8

Find the period of $\sin (x - \frac{π}{4})$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

Add $2 π$ to every negative angle to get positive angles.

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Step 9.1

Add $2 π$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + 2 π$

Step 9.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$2 π \cdot \frac{12}{12} - \frac{π}{12}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $2 π$ and $\frac{12}{12}$ .

$\frac{2 π \cdot 12}{12} - \frac{π}{12}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 12 - π}{12}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $12$ by $2$ .

$\frac{24 π - π}{12}$

Step 9.4.2

Subtract $π$ from $24 π$ .

$\frac{23 π}{12}$

Step 9.5

List the new angles.

$x = \frac{23 π}{12}$

Step 10

The period of the $\sin (x - \frac{π}{4})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{19 π}{12} + 2 π n, \frac{23 π}{12} + 2 π n$ , for any integer $n$