Trigonometry Examples

$\sin (\frac{π}{4} x) = 0$

Step 1

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

$\frac{π}{4} x = \arcsin (0)$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{π}{4}$ and $x$ .

$\frac{π x}{4} = \arcsin (0)$

Step 3

Simplify the right side.

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Step 3.1

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

$\frac{π x}{4} = 0$

Step 4

Set the numerator equal to zero.

$π x = 0$

Step 5

Divide each term in $π x = 0$ by $π$ and simplify.

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Step 5.1

Divide each term in $π x = 0$ by $π$ .

$\frac{π x}{π} = \frac{0}{π}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $π$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{0}{π}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{π}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $π$ .

$x = 0$

Step 6

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.

$\frac{π x}{4} = π - 0$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $\frac{4}{π}$ .

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

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Simplify $\frac{4}{π} \cdot \frac{π x}{4}$ .

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Step 7.2.1.1.1

Cancel the common factor of $4$ .

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Step 7.2.1.1.1.1

Cancel the common factor.

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

Step 7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{4}{π} (π - 0)$

Step 7.2.1.1.2

Cancel the common factor of $π$ .

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Step 7.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{4}{π} (π - 0)$

Step 7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{4}{π} (π - 0)$

Step 7.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{π} (π - 0)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $\frac{4}{π} (π - 0)$ .

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Step 7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{4}{π} π$

Step 7.2.2.1.2

Cancel the common factor of $π$ .

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Step 7.2.2.1.2.1

Cancel the common factor.

$x = \frac{4}{π} π$

Step 7.2.2.1.2.2

Rewrite the expression.

$x = 4$

Step 8

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

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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 $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Step 8.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 8.5

Cancel the common factor of $π$ .

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Step 8.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 8.5.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 8.5.3

Rewrite the expression.

$2 \cdot 4$

Step 8.6

Multiply $2$ by $4$ .

$8$

Step 9

The period of the $\sin (\frac{π}{4} x)$ function is $8$ so values will repeat every $8$ radians in both directions.

$x = 8 n, 4 + 8 n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = 4 n$ , for any integer $n$