Trigonometry Examples

$\sin^{2} (4 x) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$\sin^{2} (4 x) = 1$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (4 x) = \pm \sqrt{1}$

Step 3

Any root of $1$ is $1$ .

$\sin (4 x) = \pm 1$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$\sin (4 x) = 1$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (4 x) = - 1$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (4 x) = 1, - 1$

Step 5

Set up each of the solutions to solve for $x$ .

$\sin (4 x) = 1$

$\sin (4 x) = - 1$

Step 6

Solve for $x$ in $\sin (4 x) = 1$ .

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Step 6.1

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

$4 x = \arcsin (1)$

Step 6.2

Simplify the right side.

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Step 6.2.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$4 x = \frac{π}{2}$

Step 6.3

Divide each term in $4 x = \frac{π}{2}$ by $4$ and simplify.

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Step 6.3.1

Divide each term in $4 x = \frac{π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $4$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{4}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{4}$

Step 6.3.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

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Step 6.3.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Step 6.3.3.2.2

Multiply $2$ by $4$ .

$x = \frac{π}{8}$

Step 6.4

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.

$4 x = π - \frac{π}{2}$

Step 6.5

Solve for $x$ .

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Step 6.5.1

Simplify.

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Step 6.5.1.1

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

$4 x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.5.1.2

Combine $π$ and $\frac{2}{2}$ .

$4 x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 6.5.1.3

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 2 - π}{2}$

Step 6.5.1.4

Subtract $π$ from $π \cdot 2$ .

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Step 6.5.1.4.1

Reorder $π$ and $2$ .

$4 x = \frac{2 \cdot π - π}{2}$

Step 6.5.1.4.2

Subtract $π$ from $2 \cdot π$ .

$4 x = \frac{π}{2}$

Step 6.5.2

Divide each term in $4 x = \frac{π}{2}$ by $4$ and simplify.

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Step 6.5.2.1

Divide each term in $4 x = \frac{π}{2}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 6.5.2.2

Simplify the left side.

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Step 6.5.2.2.1

Cancel the common factor of $4$ .

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Step 6.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{2}}{4}$

Step 6.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{4}$

Step 6.5.2.3

Simplify the right side.

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Step 6.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{4}$

Step 6.5.2.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{4}$ .

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Step 6.5.2.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{4}$ .

$x = \frac{π}{2 \cdot 4}$

Step 6.5.2.3.2.2

Multiply $2$ by $4$ .

$x = \frac{π}{8}$

Step 6.6

Find the period of $\sin (4 x)$ .

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Step 6.6.1

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

$\frac{2 π}{| b |}$

Step 6.6.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 6.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 6.6.4

Cancel the common factor of $2$ and $4$ .

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Step 6.6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 6.6.4.2

Cancel the common factors.

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Step 6.6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 6.6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 6.6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 6.7

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

$x = \frac{π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 7

Solve for $x$ in $\sin (4 x) = - 1$ .

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Step 7.1

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

$4 x = \arcsin (- 1)$

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$4 x = - \frac{π}{2}$

Step 7.3

Divide each term in $4 x = - \frac{π}{2}$ by $4$ and simplify.

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Step 7.3.1

Divide each term in $4 x = - \frac{π}{2}$ by $4$ .

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

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $4$ .

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Step 7.3.2.1.1

Cancel the common factor.

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

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{4}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{2} \cdot \frac{1}{4}$

Step 7.3.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{4}$ .

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Step 7.3.3.2.1

Multiply $\frac{1}{4}$ by $\frac{π}{2}$ .

$x = - \frac{π}{4 \cdot 2}$

Step 7.3.3.2.2

Multiply $4$ by $2$ .

$x = - \frac{π}{8}$

Step 7.4

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.

$4 x = 2 π + \frac{π}{2} + π$

Step 7.5

Simplify the expression to find the second solution.

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Step 7.5.1

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

$4 x = 2 π + \frac{π}{2} + π - 2 π$

Step 7.5.2

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

$4 x = \frac{3 π}{2}$

Step 7.5.3

Divide each term in $4 x = \frac{3 π}{2}$ by $4$ and simplify.

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Step 7.5.3.1

Divide each term in $4 x = \frac{3 π}{2}$ by $4$ .

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

Step 7.5.3.2

Simplify the left side.

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Step 7.5.3.2.1

Cancel the common factor of $4$ .

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Step 7.5.3.2.1.1

Cancel the common factor.

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

Step 7.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{4}$

Step 7.5.3.3

Simplify the right side.

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Step 7.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{4}$

Step 7.5.3.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{4}$ .

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Step 7.5.3.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{4}$ .

$x = \frac{3 π}{2 \cdot 4}$

Step 7.5.3.3.2.2

Multiply $2$ by $4$ .

$x = \frac{3 π}{8}$

Step 7.6

Find the period of $\sin (4 x)$ .

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Step 7.6.1

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

$\frac{2 π}{| b |}$

Step 7.6.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 7.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 7.6.4

Cancel the common factor of $2$ and $4$ .

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Step 7.6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 7.6.4.2

Cancel the common factors.

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Step 7.6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 7.6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 7.6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 7.7

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 7.7.1

Add $\frac{π}{2}$ to $- \frac{π}{8}$ to find the positive angle.

$- \frac{π}{8} + \frac{π}{2}$

Step 7.7.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{π}{2} \cdot \frac{4}{4} - \frac{π}{8}$

Step 7.7.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.7.3.1

Multiply $\frac{π}{2}$ by $\frac{4}{4}$ .

$\frac{π \cdot 4}{2 \cdot 4} - \frac{π}{8}$

Step 7.7.3.2

Multiply $2$ by $4$ .

$\frac{π \cdot 4}{8} - \frac{π}{8}$

Step 7.7.4

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{8}$

Step 7.7.5

Simplify the numerator.

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Step 7.7.5.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{8}$

Step 7.7.5.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{8}$

Step 7.7.6

List the new angles.

$x = \frac{3 π}{8}$

Step 7.8

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

$x = \frac{3 π}{8} + \frac{π n}{2}, \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 8

List all of the solutions.

$x = \frac{π}{8} + \frac{π n}{2}, \frac{3 π}{8} + \frac{π n}{2}, \frac{3 π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 9

Consolidate the answers.

$x = \frac{π}{8} + \frac{π n}{4}$ , for any integer $n$