Trigonometry Examples

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

Add $1$ to both sides of the equation.

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

Divide each term by $4$ and simplify.

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Divide each term in $4 \sin^{2} (x) = 1$ by $4$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

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

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

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

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

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

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Simplify the right side of the equation.

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Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

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

Any root of $1$ is $1$ .

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

Simplify the denominator.

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

$\sin (x) = \pm \frac{1}{\sqrt{2^{2}}}$

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

$\sin (x) = \pm \frac{1}{2}$

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

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

$\sin (x) = \frac{1}{2}$

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

$\sin (x) = - \frac{1}{2}$

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

$\sin (x) = \frac{1}{2}, - \frac{1}{2}$

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

$\sin (x) = \frac{1}{2}$

$\sin (x) = - \frac{1}{2}$

Set up the equation to solve for $x$ .

$\sin (x) = \frac{1}{2}$

Solve the equation for $x$ .

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

$x = \arcsin (\frac{1}{2})$

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

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.

$x = π - \frac{π}{6}$

Simplify $π - \frac{π}{6}$ .

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To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Combine fractions.

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Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Simplify the numerator.

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Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

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

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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.

$\frac{2 π}{| 1 |}$

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\sin (x) = - \frac{1}{2}$

Solve the equation for $x$ .

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

$x = \arcsin (- \frac{1}{2})$

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$x = - \frac{π}{6}$

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.

$x = 2 π + \frac{π}{6} + π$

Simplify the expression to find the second solution.

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Simplify $2 π + \frac{π}{6} + π$ .

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Find the common denominator.

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Write $2 π$ as a fraction with denominator $1$ .

$x = \frac{2 π}{1} + \frac{π}{6} + π$

Multiply $\frac{2 π}{1}$ by $\frac{6}{6}$ .

$x = \frac{2 π}{1} \cdot \frac{6}{6} + \frac{π}{6} + π$

Multiply $\frac{2 π}{1}$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + π$

Write $π$ as a fraction with denominator $1$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + \frac{π}{1}$

Multiply $\frac{π}{1}$ by $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + \frac{π}{1} \cdot \frac{6}{6}$

Multiply $\frac{π}{1}$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + \frac{π \cdot 6}{6}$

Combine fractions.

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Combine fractions with similar denominators.

$x = \frac{2 π \cdot 6 + π + π \cdot 6}{6}$

Simplify the expression.

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Multiply $6$ by $2$ .

$x = \frac{12 π + π + π \cdot 6}{6}$

Move $6$ to the left of $π$ .

$x = \frac{12 π + π + 6 π}{6}$

Simplify the numerator.

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Add $12 π$ and $π$ .

$x = \frac{13 π + 6 π}{6}$

Add $13 π$ and $6 π$ .

$x = \frac{19 π}{6}$

Subtract $2 π$ from $\frac{19 π}{6}$ .

$x = \frac{19 π}{6} - 2 π$

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $\frac{19 π}{6}$ .

$x = \frac{7 π}{6}$

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

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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.

$\frac{2 π}{| 1 |}$

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 $1$ .

$2 π$

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

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Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Combine fractions.

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Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

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Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

List the new angles.

$x = \frac{11 π}{6}$

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

List all of the results found in the previous steps.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate the answers.

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Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Consolidate $\frac{5 π}{6} + 2 π n$ and $\frac{11 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

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