Trigonometry Examples

$4 \sin^{2} (x) - 3 = 0$ , $[0, 2 π)$

Step 1

Add $3$ to both sides of the equation.

$4 \sin^{2} (x) = 3$

Step 2

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

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Step 2.1

Divide each term in $4 \sin^{2} (x) = 3$ by $4$ .

$\frac{4 \sin^{2} (x)}{4} = \frac{3}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 \sin^{2} (x)}{4} = \frac{3}{4}$

Step 2.2.1.2

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{3}{4}$

Step 3

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

$\sin (x) = \pm \sqrt{\frac{3}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{3}{4}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{\sqrt{4}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Rewrite $4$ as $2^{2}$ .

$\sin (x) = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm \frac{\sqrt{3}}{2}$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

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

Step 7

Solve for $x$ in $\sin (x) = \frac{\sqrt{3}}{2}$ .

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Step 7.1

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

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

Step 7.2

Simplify the right side.

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Step 7.2.1

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

$x = \frac{π}{3}$

Step 7.3

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.

$x = π - \frac{π}{3}$

Step 7.4

Simplify $π - \frac{π}{3}$ .

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Step 7.4.1

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

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

Step 7.4.2

Combine fractions.

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Step 7.4.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 7.4.3

Simplify the numerator.

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Step 7.4.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 7.4.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 7.5

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

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Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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

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

Step 8

Solve for $x$ in $\sin (x) = - \frac{\sqrt{3}}{2}$ .

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Step 8.1

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

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

Step 8.2

Simplify the right side.

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Step 8.2.1

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

$x = - \frac{π}{3}$

Step 8.3

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.

$x = 2 π + \frac{π}{3} + π$

Step 8.4

Simplify the expression to find the second solution.

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Step 8.4.1

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

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 8.4.2

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

$x = \frac{4 π}{3}$

Step 8.5

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

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Step 8.5.1

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

$\frac{2 π}{| b |}$

Step 8.5.2

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

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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

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Step 8.6.1

Add $2 π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + 2 π$

Step 8.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 8.6.3

Combine fractions.

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Step 8.6.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 8.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 8.6.4

Simplify the numerator.

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Step 8.6.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 8.6.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 8.6.5

List the new angles.

$x = \frac{5 π}{3}$

Step 8.7

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

$x = \frac{4 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 9

List all of the solutions.

$x = \frac{π}{3} + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $\frac{π}{3} + 2 π n$ and $\frac{4 π}{3} + 2 π n$ to $\frac{π}{3} + π n$ .

$x = \frac{π}{3} + π n, \frac{2 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 10.2

Consolidate $\frac{2 π}{3} + 2 π n$ and $\frac{5 π}{3} + 2 π n$ to $\frac{2 π}{3} + π n$ .

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

Step 11

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 11.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 11.1.1

Plug in $0$ for $n$ .

$\frac{π}{3} + π (0)$

Step 11.1.2

Simplify.

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Step 11.1.2.1

Multiply $π$ by $0$ .

$\frac{π}{3} + 0$

Step 11.1.2.2

Add $\frac{π}{3}$ and $0$ .

$\frac{π}{3}$

Step 11.1.3

The interval $[0, 2 π)$ contains $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 11.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 11.2.1

Plug in $0$ for $n$ .

$\frac{2 π}{3} + π (0)$

Step 11.2.2

Simplify.

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Step 11.2.2.1

Multiply $π$ by $0$ .

$\frac{2 π}{3} + 0$

Step 11.2.2.2

Add $\frac{2 π}{3}$ and $0$ .

$\frac{2 π}{3}$

Step 11.2.3

The interval $[0, 2 π)$ contains $\frac{2 π}{3}$ .

$x = \frac{π}{3}, \frac{2 π}{3}$

Step 11.3

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 11.3.1

Plug in $1$ for $n$ .

$\frac{π}{3} + π (1)$

Step 11.3.2

Simplify.

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Step 11.3.2.1

Multiply $π$ by $1$ .

$\frac{π}{3} + π$

Step 11.3.2.2

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

$\frac{π}{3} + π \cdot \frac{3}{3}$

Step 11.3.2.3

Combine fractions.

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Step 11.3.2.3.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{π}{3} + \frac{π \cdot 3}{3}$

Step 11.3.2.3.2

Combine the numerators over the common denominator.

$\frac{π + π \cdot 3}{3}$

Step 11.3.2.4

Simplify the numerator.

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Step 11.3.2.4.1

Move $3$ to the left of $π$ .

$\frac{π + 3 \cdot π}{3}$

Step 11.3.2.4.2

Add $π$ and $3 π$ .

$\frac{4 π}{3}$

Step 11.3.3

The interval $[0, 2 π)$ contains $\frac{4 π}{3}$ .

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

Step 11.4

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 11.4.1

Plug in $1$ for $n$ .

$\frac{2 π}{3} + π (1)$

Step 11.4.2

Simplify.

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Step 11.4.2.1

Multiply $π$ by $1$ .

$\frac{2 π}{3} + π$

Step 11.4.2.2

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

$\frac{2 π}{3} + π \cdot \frac{3}{3}$

Step 11.4.2.3

Combine fractions.

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Step 11.4.2.3.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{2 π}{3} + \frac{π \cdot 3}{3}$

Step 11.4.2.3.2

Combine the numerators over the common denominator.

$\frac{2 π + π \cdot 3}{3}$

Step 11.4.2.4

Simplify the numerator.

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Step 11.4.2.4.1

Move $3$ to the left of $π$ .

$\frac{2 π + 3 \cdot π}{3}$

Step 11.4.2.4.2

Add $2 π$ and $3 π$ .

$\frac{5 π}{3}$

Step 11.4.3

The interval $[0, 2 π)$ contains $\frac{5 π}{3}$ .

$x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$