Trigonometry Examples

$\sin (x) - \sqrt{3 - 3 \sin^{2} (x)} = 0$

Step 1

Subtract $\sin (x)$ from both sides of the equation.

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

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 3 \sin^{2} (x)}$ as ${(3 - 3 \sin^{2} (x))}^{\frac{1}{2}}$ .

${(- {(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2} = {(- \sin (x))}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- {(3 - 3 \sin^{2} (x))}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2} = {(- \sin (x))}^{2}$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2} = {(- \sin (x))}^{2}$

Step 3.2.1.3

Multiply ${({(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2} = {(- \sin (x))}^{2}$

Step 3.2.1.4

Multiply the exponents in ${({(3 - 3 \sin^{2} (x))}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 - 3 \sin^{2} (x))}^{\frac{1}{2} \cdot 2} = {(- \sin (x))}^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

${(3 - 3 \sin^{2} (x))}^{\frac{1}{2} \cdot 2} = {(- \sin (x))}^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

${(3 - 3 \sin^{2} (x))}^{1} = {(- \sin (x))}^{2}$

Step 3.2.1.5

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- \sin (x))}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $- \sin (x)$ .

$3 - 3 \sin^{2} (x) = {(- 1)}^{2} \sin^{2} (x)$

Step 3.3.1.2

Raise $- 1$ to the power of $2$ .

$3 - 3 \sin^{2} (x) = 1 \sin^{2} (x)$

Step 3.3.1.3

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

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

Step 4

Solve for $\sin (x)$ .

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Step 4.1

Move all terms containing $\sin (x)$ to the left side of the equation.

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Step 4.1.1

Subtract $\sin^{2} (x)$ from both sides of the equation.

$3 - 3 \sin^{2} (x) - \sin^{2} (x) = 0$

Step 4.1.2

Subtract $\sin^{2} (x)$ from $- 3 \sin^{2} (x)$ .

$3 - 4 \sin^{2} (x) = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 4$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

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

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Dividing two negative values results in a positive value.

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

Step 4.4

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 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Simplify the denominator.

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Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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 5

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

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the right side.

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Step 6.2.1

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

$x = \frac{π}{3}$

Step 6.3

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 6.4

Simplify $π - \frac{π}{3}$ .

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Step 6.4.1

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 6.4.2

Combine fractions.

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Step 6.4.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 6.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 6.4.3

Simplify the numerator.

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Step 6.4.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 6.4.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 6.5

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

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Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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 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 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 7.4

Simplify the expression to find the second solution.

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Step 7.4.1

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

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 7.4.2

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 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

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

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Step 7.6.1

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

$- \frac{π}{3} + 2 π$

Step 7.6.2

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 7.6.3

Combine fractions.

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Step 7.6.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 7.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 7.6.4

Simplify the numerator.

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Step 7.6.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 7.6.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 7.6.5

List the new angles.

$x = \frac{5 π}{3}$

Step 7.7

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 8

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 9

Consolidate the solutions.

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Step 9.1

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 9.2

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 10

Exclude the solutions that do not make $\sin (x) - \sqrt{3 - 3 \sin^{2} (x)} = 0$ true.

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