Solve for x f(x)*(2sin(x-pi/4))=0

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Trigonometry

$f (x) \cdot (2 \sin (x - \frac{π}{4})) = 0$

Divide each term by $2 f (x)$ and simplify.

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Divide each term in $f (x) \cdot (2 \sin (x - \frac{π}{4})) = 0$ by $2 f (x)$ .

$\frac{f (x) \cdot (2 \sin (x - \frac{π}{4}))}{2 f (x)} = \frac{0}{2 f (x)}$

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{f (x) \cdot (2 \sin (x - \frac{π}{4}))}{2 f x} = \frac{0}{2 f (x)}$

Rewrite the expression.

$\frac{f (x) \cdot (\sin (x - \frac{π}{4}))}{f (x)} = \frac{0}{2 f (x)}$

Reorder factors in $\frac{f (x) \sin (x - \frac{π}{4})}{f x}$ .

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = \frac{0}{2 f (x)}$

Simplify $\frac{0}{2 f (x)}$ .

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Cancel the common factor of $0$ and $2$ .

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Factor $2$ out of $0$ .

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = \frac{2 (0)}{2 f (x)}$

Cancel the common factors.

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Factor $2$ out of $2 f (x)$ .

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = \frac{2 (0)}{2 (f (x))}$

Cancel the common factor.

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = \frac{2 \cdot 0}{2 (f (x))}$

Rewrite the expression.

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = \frac{0}{f (x)}$

Divide $0$ by $f x$ .

$\frac{\sin (x - \frac{π}{4}) f (x)}{f x} = 0$

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

$x - \frac{π}{4} = \arcsin (0)$

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

$x - \frac{π}{4} = 0$

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

$x = \frac{π}{4}$

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.

$x - \frac{π}{4} = π - 0$

Simplify the expression to find the second solution.

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Subtract $0$ from $π$ .

$x - \frac{π}{4} = π$

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

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

$x = π + \frac{π}{4}$

Simplify the right side of the equation.

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{π}{1} \cdot \frac{4}{4} + \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{π \cdot 4}{1 \cdot 4} + \frac{π}{4}$

Multiply $4$ by $1$ .

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

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Simplify the numerator.

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Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Find the period.

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

Solve the equation.

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$x = \frac{π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Consolidate the answers.

$x = \frac{π}{4} + π n$ , for any integer $n$

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