Trigonometry Examples

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

Step 1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 2

Add $\frac{\sin (x)}{\cos (x)}$ to both sides of the equation.

$\frac{\sin (2 x)}{1 - 2 \sin^{2} (x)} + \frac{\sin (x)}{\cos (x)} = 0$

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the sine double-angle identity.

$\frac{2 \sin (x) \cos (x)}{1 - 2 \sin^{2} (x)} + \frac{\sin (x)}{\cos (x)} = 0$

Step 3.1.1.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{2 \sin (x) \cos (x)}{1 - 2 \sin^{2} (x)} + \tan (x) = 0$

Step 3.1.2

To write $\tan (x)$ as a fraction with a common denominator, multiply by $\frac{1 - 2 \sin^{2} (x)}{1 - 2 \sin^{2} (x)}$ .

$\frac{2 \sin (x) \cos (x)}{1 - 2 \sin^{2} (x)} + \frac{\tan (x) (1 - 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.3

Combine the numerators over the common denominator.

$\frac{2 \sin (x) \cos (x) + \tan (x) (1 - 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Apply the sine double-angle identity.

$\frac{\sin (2 x) + \tan (x) (1 - 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.2

Apply the cosine double-angle identity.

$\frac{\sin (2 x) + \tan (x) \cos (2 x)}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.3

Simplify each term.

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Step 3.1.4.3.1

Apply the sine double-angle identity.

$\frac{2 \sin (x) \cos (x) + \tan (x) \cos (2 x)}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.3.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$\frac{2 \sin (x) \cos (x) + \tan (x) (1 - 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.3.3

Apply the distributive property.

$\frac{2 \sin (x) \cos (x) + \tan (x) \cdot 1 + \tan (x) (- 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.3.4

Multiply $\tan (x)$ by $1$ .

$\frac{2 \sin (x) \cos (x) + \tan (x) + \tan (x) (- 2 \sin^{2} (x))}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.3.5

Rewrite using the commutative property of multiplication.

$\frac{2 \sin (x) \cos (x) + \tan (x) - 2 \tan (x) \sin^{2} (x)}{1 - 2 \sin^{2} (x)} = 0$

Step 3.1.4.4

Apply the sine double-angle identity.

$\frac{\sin (2 x) + \tan (x) - 2 \tan (x) \sin^{2} (x)}{1 - 2 \sin^{2} (x)} = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Set the numerator equal to zero.

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

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - 2 \tan (x) \sin^{2} (x) = 0$

Step 4.2.1.1.2

Rewrite $\tan (x)$ in terms of sines and cosines.

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - 2 \frac{\sin (x)}{\cos (x)} \sin^{2} (x) = 0$

Step 4.2.1.1.3

Combine $- 2$ and $\frac{\sin (x)}{\cos (x)}$ .

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} + \frac{- 2 \sin (x)}{\cos (x)} \sin^{2} (x) = 0$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{2 \sin (x)}{\cos (x)} \sin^{2} (x) = 0$

Step 4.2.1.1.5

Multiply $- \frac{2 \sin (x)}{\cos (x)} \sin^{2} (x)$ .

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Step 4.2.1.1.5.1

Combine $\sin^{2} (x)$ and $\frac{2 \sin (x)}{\cos (x)}$ .

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{\sin^{2} (x) (2 \sin (x))}{\cos (x)} = 0$

Step 4.2.1.1.5.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ by adding the exponents.

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Step 4.2.1.1.5.2.1

Move $\sin (x)$ .

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{\sin (x) \sin^{2} (x) \cdot 2}{\cos (x)} = 0$

Step 4.2.1.1.5.2.2

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

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Step 4.2.1.1.5.2.2.1

Raise $\sin (x)$ to the power of $1$ .

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{\sin^{1} (x) \sin^{2} (x) \cdot 2}{\cos (x)} = 0$

Step 4.2.1.1.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{{\sin (x)}^{1 + 2} \cdot 2}{\cos (x)} = 0$

Step 4.2.1.1.5.2.3

Add $1$ and $2$ .

$\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{\sin^{3} (x) \cdot 2}{\cos (x)} = 0$

Step 4.2.1.1.6

Move $2$ to the left of $\sin^{3} (x)$ .

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

Step 4.2.2

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\sin (2 x) + \frac{\sin (x)}{\cos (x)} - \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \cdot 0$

Step 4.2.3

Apply the distributive property.

$\cos (x) \sin (2 x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \cdot 0$

Step 4.2.4

Simplify.

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Step 4.2.4.1

Cancel the common factor of $\cos (x)$ .

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Step 4.2.4.1.1

Cancel the common factor.

$\cos (x) \sin (2 x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \cdot 0$

Step 4.2.4.1.2

Rewrite the expression.

$\cos (x) \sin (2 x) + \sin (x) + \cos (x) (- \frac{2 \sin^{3} (x)}{\cos (x)}) = \cos (x) \cdot 0$

Step 4.2.4.2

Rewrite using the commutative property of multiplication.

$\cos (x) \sin (2 x) + \sin (x) - \cos (x) \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \cdot 0$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Cancel the common factor of $\cos (x)$ .

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Step 4.2.5.1.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\cos (x) \sin (2 x) + \sin (x) + \cos (x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \cdot 0$

Step 4.2.5.1.2

Cancel the common factor.

$\cos (x) \sin (2 x) + \sin (x) + \cos (x) \cdot - 1 \frac{2 \sin^{3} (x)}{\cos (x)} = \cos (x) \cdot 0$

Step 4.2.5.1.3

Rewrite the expression.

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

Step 4.2.5.2

Multiply $2$ by $- 1$ .

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

Step 4.2.6

Multiply $\cos (x)$ by $0$ .

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

Step 4.2.7

Simplify each term.

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Step 4.2.7.1

Apply the sine double-angle identity.

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

Step 4.2.7.2

Rewrite using the commutative property of multiplication.

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

Step 4.2.7.3

Multiply $2 \cos (x) \sin (x) \cos (x)$ .

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Step 4.2.7.3.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.2.7.3.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.2.7.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 {\cos (x)}^{1 + 1} \sin (x) + \sin (x) - 2 \sin^{3} (x) = 0$

Step 4.2.7.3.4

Add $1$ and $1$ .

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

Step 4.2.8

Factor $\sin (x)$ out of $2 \cos^{2} (x) \sin (x) + \sin (x) - 2 \sin^{3} (x)$ .

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Step 4.2.8.1

Factor $\sin (x)$ out of $2 \cos^{2} (x) \sin (x)$ .

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

Step 4.2.8.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.2.8.3

Factor $\sin (x)$ out of $\sin^{1} (x)$ .

$\sin (x) (2 \cos^{2} (x)) + \sin (x) \cdot 1 - 2 \sin^{3} (x) = 0$

Step 4.2.8.4

Factor $\sin (x)$ out of $- 2 \sin^{3} (x)$ .

$\sin (x) (2 \cos^{2} (x)) + \sin (x) \cdot 1 + \sin (x) (- 2 \sin^{2} (x)) = 0$

Step 4.2.8.5

Factor $\sin (x)$ out of $\sin (x) (2 \cos^{2} (x)) + \sin (x) \cdot 1$ .

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

Step 4.2.8.6

Factor $\sin (x)$ out of $\sin (x) (2 \cos^{2} (x) + 1) + \sin (x) (- 2 \sin^{2} (x))$ .

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

Step 4.2.9

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (x) = 0$

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

Step 4.2.10

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 4.2.10.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 4.2.10.2

Solve $\sin (x) = 0$ for $x$ .

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Step 4.2.10.2.1

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

$x = \arcsin (0)$

Step 4.2.10.2.2

Simplify the right side.

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Step 4.2.10.2.2.1

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

$x = 0$

Step 4.2.10.2.3

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.

$x = π - 0$

Step 4.2.10.2.4

Subtract $0$ from $π$ .

$x = π$

Step 4.2.10.2.5

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

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Step 4.2.10.2.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.10.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.10.2.5.3

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

$\frac{2 π}{1}$

Step 4.2.10.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.10.2.6

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 4.2.11

Set $2 \cos^{2} (x) + 1 - 2 \sin^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 4.2.11.1

Set $2 \cos^{2} (x) + 1 - 2 \sin^{2} (x)$ equal to $0$ .

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

Step 4.2.11.2

Solve $2 \cos^{2} (x) + 1 - 2 \sin^{2} (x) = 0$ for $x$ .

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Step 4.2.11.2.1

Replace the $2 \cos^{2} (x)$ with $2 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 4.2.11.2.2

Simplify each term.

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Step 4.2.11.2.2.1

Apply the distributive property.

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

Step 4.2.11.2.2.2

Multiply $2$ by $1$ .

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

Step 4.2.11.2.2.3

Multiply $- 1$ by $2$ .

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

Step 4.2.11.2.3

Simplify by adding terms.

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Step 4.2.11.2.3.1

Add $2$ and $1$ .

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

Step 4.2.11.2.3.2

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

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

Step 4.2.11.2.4

Subtract $3$ from both sides of the equation.

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

Step 4.2.11.2.5

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

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Step 4.2.11.2.5.1

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

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

Step 4.2.11.2.5.2

Simplify the left side.

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Step 4.2.11.2.5.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.11.2.5.2.1.1

Cancel the common factor.

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

Step 4.2.11.2.5.2.1.2

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

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

Step 4.2.11.2.5.3

Simplify the right side.

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Step 4.2.11.2.5.3.1

Dividing two negative values results in a positive value.

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

Step 4.2.11.2.6

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 4.2.11.2.7

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

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Step 4.2.11.2.7.1

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

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

Step 4.2.11.2.7.2

Simplify the denominator.

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Step 4.2.11.2.7.2.1

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

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

Step 4.2.11.2.7.2.2

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

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

Step 4.2.11.2.8

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

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Step 4.2.11.2.8.1

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

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

Step 4.2.11.2.8.2

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

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

Step 4.2.11.2.8.3

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 4.2.11.2.9

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

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

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

Step 4.2.11.2.10

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

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Step 4.2.11.2.10.1

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

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

Step 4.2.11.2.10.2

Simplify the right side.

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Step 4.2.11.2.10.2.1

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

$x = \frac{π}{3}$

Step 4.2.11.2.10.3

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 4.2.11.2.10.4

Simplify $π - \frac{π}{3}$ .

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Step 4.2.11.2.10.4.1

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 4.2.11.2.10.4.2

Combine fractions.

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Step 4.2.11.2.10.4.2.1

Combine $π$ and $\frac{3}{3}$ .

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

Step 4.2.11.2.10.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Step 4.2.11.2.10.4.3

Simplify the numerator.

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Step 4.2.11.2.10.4.3.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Step 4.2.11.2.10.4.3.2

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 4.2.11.2.10.5

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

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Step 4.2.11.2.10.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.11.2.10.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.11.2.10.5.3

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

$\frac{2 π}{1}$

Step 4.2.11.2.10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.11.2.10.6

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 4.2.11.2.11

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

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Step 4.2.11.2.11.1

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

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

Step 4.2.11.2.11.2

Simplify the right side.

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Step 4.2.11.2.11.2.1

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

$x = - \frac{π}{3}$

Step 4.2.11.2.11.3

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 4.2.11.2.11.4

Simplify the expression to find the second solution.

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Step 4.2.11.2.11.4.1

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

$x = 2 π + \frac{π}{3} + π - 2 π$

Step 4.2.11.2.11.4.2

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 4.2.11.2.11.5

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

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Step 4.2.11.2.11.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.11.2.11.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.11.2.11.5.3

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

$\frac{2 π}{1}$

Step 4.2.11.2.11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.11.2.11.6

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

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Step 4.2.11.2.11.6.1

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

$- \frac{π}{3} + 2 π$

Step 4.2.11.2.11.6.2

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 4.2.11.2.11.6.3

Combine fractions.

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Step 4.2.11.2.11.6.3.1

Combine $2 π$ and $\frac{3}{3}$ .

$\frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 4.2.11.2.11.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 3 - π}{3}$

Step 4.2.11.2.11.6.4

Simplify the numerator.

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Step 4.2.11.2.11.6.4.1

Multiply $3$ by $2$ .

$\frac{6 π - π}{3}$

Step 4.2.11.2.11.6.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{3}$

Step 4.2.11.2.11.6.5

List the new angles.

$x = \frac{5 π}{3}$

Step 4.2.11.2.11.7

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 4.2.11.2.12

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 4.2.11.2.13

Consolidate the solutions.

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Step 4.2.11.2.13.1

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 4.2.11.2.13.2

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 4.2.12

The final solution is all the values that make $\sin (x) (2 \cos^{2} (x) + 1 - 2 \sin^{2} (x)) = 0$ true.

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

Step 5

Consolidate the answers.

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Step 5.1

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

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

Step 5.2

Consolidate the answers.

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