Precalculus Examples

$6 + 6 \sin (x) = 4 \cos^{2} (x)$

Step 1

Subtract $4 \cos^{2} (x)$ from both sides of the equation.

$6 + 6 \sin (x) - 4 \cos^{2} (x) = 0$

Step 2

Factor $2$ out of $6 + 6 \sin (x) - 4 \cos^{2} (x)$ .

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Step 2.1

Factor $2$ out of $6$ .

$2 (3) + 6 \sin (x) - 4 \cos^{2} (x) = 0$

Step 2.2

Factor $2$ out of $6 \sin (x)$ .

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

Step 2.3

Factor $2$ out of $- 4 \cos^{2} (x)$ .

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

Step 2.4

Factor $2$ out of $2 (3) + 2 (3 \sin (x))$ .

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

Step 2.5

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

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

Step 3

Divide each term in the equation by $\cos (x)$ .

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

Step 4

Replace $\cos (x)$ with an equivalent expression in the numerator.

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

Step 5

Remove parentheses.

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

Multiply $2$ by $3$ .

$(6 + 2 (3 \sin (x)) + 2 (- 2 \cos^{2} (x))) \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 7.2

Multiply $3$ by $2$ .

$(6 + 6 \sin (x) + 2 (- 2 \cos^{2} (x))) \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 7.3

Multiply $- 2$ by $2$ .

$(6 + 6 \sin (x) - 4 \cos^{2} (x)) \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 8

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

$(6 + 6 \sin (x) - 4 \cos^{2} (x)) \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9

Apply the distributive property.

$6 (\frac{1}{\cos (x)}) + 6 \sin (x) (\frac{1}{\cos (x)}) - 4 \cos^{2} (x) \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 10

Simplify.

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Step 10.1

Combine $6$ and $\frac{1}{\cos (x)}$ .

$\frac{6}{\cos (x)} + 6 \sin (x) (\frac{1}{\cos (x)}) - 4 \cos^{2} (x) \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 10.2

Multiply $6 \sin (x) \frac{1}{\cos (x)}$ .

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Step 10.2.1

Combine $\frac{1}{\cos (x)}$ and $6$ .

$\frac{6}{\cos (x)} + \frac{6}{\cos (x)} \cdot \sin (x) - 4 \cos^{2} (x) \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 10.2.2

Combine $\frac{6}{\cos (x)}$ and $\sin (x)$ .

$\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} - 4 \cos^{2} (x) \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 10.3

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

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Step 10.3.1

Factor $\cos (x)$ out of $- 4 \cos^{2} (x)$ .

$\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) (\frac{1}{\cos (x)}) = \frac{0}{\cos (x)}$

Step 10.3.2

Cancel the common factor.

$\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) (\frac{1}{\cos (x)}) = \frac{0}{\cos (x)}$

Step 10.3.3

Rewrite the expression.

$\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11

Simplify each term.

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Step 11.1

Separate fractions.

$\frac{6}{1} \cdot \frac{1}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{6}{1} \cdot \sec (x) + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11.3

Divide $6$ by $1$ .

$6 \sec (x) + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11.4

Separate fractions.

$6 \sec (x) + \frac{6}{1} \cdot \frac{\sin (x)}{\cos (x)} - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11.5

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

$6 \sec (x) + \frac{6}{1} \cdot \tan (x) - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 11.6

Divide $6$ by $1$ .

$6 \sec (x) + 6 \tan (x) - 4 \cos (x) = \frac{0}{\cos (x)}$

Step 12

Separate fractions.

$6 \sec (x) + 6 \tan (x) - 4 \cos (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 13

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$6 \sec (x) + 6 \tan (x) - 4 \cos (x) = \frac{0}{1} \cdot \sec (x)$

Step 14

Divide $0$ by $1$ .

$6 \sec (x) + 6 \tan (x) - 4 \cos (x) = 0 \sec (x)$

Step 15

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

$6 \sec (x) + 6 \tan (x) - 4 \cos (x) = 0$

Step 16

Simplify the left side.

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Step 16.1

Simplify each term.

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Step 16.1.1

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

$6 \frac{1}{\cos (x)} + 6 \tan (x) - 4 \cos (x) = 0$

Step 16.1.2

Combine $6$ and $\frac{1}{\cos (x)}$ .

$\frac{6}{\cos (x)} + 6 \tan (x) - 4 \cos (x) = 0$

Step 16.1.3

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

$\frac{6}{\cos (x)} + 6 \frac{\sin (x)}{\cos (x)} - 4 \cos (x) = 0$

Step 16.1.4

Combine $6$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x) = 0$

Step 17

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

$\cos (x) (\frac{6}{\cos (x)} + \frac{6 \sin (x)}{\cos (x)} - 4 \cos (x)) = \cos (x) \cdot 0$

Step 18

Apply the distributive property.

$\cos (x) \frac{6}{\cos (x)} + \cos (x) \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) = \cos (x) \cdot 0$

Step 19

Simplify.

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Step 19.1

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

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Step 19.1.1

Cancel the common factor.

$\cos (x) \frac{6}{\cos (x)} + \cos (x) \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) = \cos (x) \cdot 0$

Step 19.1.2

Rewrite the expression.

$6 + \cos (x) \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) = \cos (x) \cdot 0$

Step 19.2

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

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Step 19.2.1

Cancel the common factor.

$6 + \cos (x) \frac{6 \sin (x)}{\cos (x)} + \cos (x) (- 4 \cos (x)) = \cos (x) \cdot 0$

Step 19.2.2

Rewrite the expression.

$6 + 6 \sin (x) + \cos (x) (- 4 \cos (x)) = \cos (x) \cdot 0$

Step 19.3

Rewrite using the commutative property of multiplication.

$6 + 6 \sin (x) - 4 \cos (x) \cos (x) = \cos (x) \cdot 0$

Step 20

Multiply $- 4 \cos (x) \cos (x)$ .

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Step 20.1

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

$6 + 6 \sin (x) - 4 (\cos^{1} (x) \cos (x)) = \cos (x) \cdot 0$

Step 20.2

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

$6 + 6 \sin (x) - 4 (\cos^{1} (x) \cos^{1} (x)) = \cos (x) \cdot 0$

Step 20.3

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

$6 + 6 \sin (x) - 4 {\cos (x)}^{1 + 1} = \cos (x) \cdot 0$

Step 20.4

Add $1$ and $1$ .

$6 + 6 \sin (x) - 4 \cos^{2} (x) = \cos (x) \cdot 0$

Step 21

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

$6 + 6 \sin (x) - 4 \cos^{2} (x) = 0$

Step 22

Replace the $- 4 \cos^{2} (x)$ with $- 4 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$6 + 6 \sin (x) - 4 (1 - \sin^{2} (x)) = 0$

Step 23

Simplify each term.

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Step 23.1

Apply the distributive property.

$6 + 6 \sin (x) - 4 \cdot 1 - 4 (- \sin^{2} (x)) = 0$

Step 23.2

Multiply $- 4$ by $1$ .

$6 + 6 \sin (x) - 4 - 4 (- \sin^{2} (x)) = 0$

Step 23.3

Multiply $- 1$ by $- 4$ .

$6 + 6 \sin (x) - 4 + 4 \sin^{2} (x) = 0$

Step 24

Subtract $4$ from $6$ .

$6 \sin (x) + 2 + 4 \sin^{2} (x) = 0$

Step 25

Reorder the polynomial.

$4 \sin^{2} (x) + 6 \sin (x) + 2 = 0$

Step 26

Substitute $u$ for $\sin (x)$ .

$4 {(u)}^{2} + 6 (u) + 2 = 0$

Step 27

Factor the left side of the equation.

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Step 27.1

Factor $2$ out of $4 u^{2} + 6 u + 2$ .

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Step 27.1.1

Factor $2$ out of $4 u^{2}$ .

$2 (2 u^{2}) + 6 u + 2 = 0$

Step 27.1.2

Factor $2$ out of $6 u$ .

$2 (2 u^{2}) + 2 (3 u) + 2 = 0$

Step 27.1.3

Factor $2$ out of $2$ .

$2 (2 u^{2}) + 2 (3 u) + 2 (1) = 0$

Step 27.1.4

Factor $2$ out of $2 (2 u^{2}) + 2 (3 u)$ .

$2 (2 u^{2} + 3 u) + 2 (1) = 0$

Step 27.1.5

Factor $2$ out of $2 (2 u^{2} + 3 u) + 2 (1)$ .

$2 (2 u^{2} + 3 u + 1) = 0$

Step 27.2

Factor.

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Step 27.2.1

Factor by grouping.

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Step 27.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

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Step 27.2.1.1.1

Factor $3$ out of $3 u$ .

$2 (2 u^{2} + 3 u + 1) = 0$

Step 27.2.1.1.2

Rewrite $3$ as $1$ plus $2$

$2 (2 u^{2} + (1 + 2) u + 1) = 0$

Step 27.2.1.1.3

Apply the distributive property.

$2 (2 u^{2} + 1 u + 2 u + 1) = 0$

Step 27.2.1.1.4

Multiply $u$ by $1$ .

$2 (2 u^{2} + u + 2 u + 1) = 0$

Step 27.2.1.2

Factor out the greatest common factor from each group.

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Step 27.2.1.2.1

Group the first two terms and the last two terms.

$2 (2 u^{2} + u + 2 u + 1) = 0$

Step 27.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 (u (2 u + 1) + 1 (2 u + 1)) = 0$

Step 27.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$2 ((2 u + 1) (u + 1)) = 0$

Step 27.2.2

Remove unnecessary parentheses.

$2 (2 u + 1) (u + 1) = 0$

Step 28

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

$2 u + 1 = 0$

$u + 1 = 0$

Step 29

Set $2 u + 1$ equal to $0$ and solve for $u$ .

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Step 29.1

Set $2 u + 1$ equal to $0$ .

$2 u + 1 = 0$

Step 29.2

Solve $2 u + 1 = 0$ for $u$ .

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Step 29.2.1

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Step 29.2.2

Divide each term in $2 u = - 1$ by $2$ and simplify.

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Step 29.2.2.1

Divide each term in $2 u = - 1$ by $2$ .

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 29.2.2.2

Simplify the left side.

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Step 29.2.2.2.1

Cancel the common factor of $2$ .

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Step 29.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 1}{2}$

Step 29.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 1}{2}$

Step 29.2.2.3

Simplify the right side.

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Step 29.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{1}{2}$

Step 30

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 30.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 30.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 31

The final solution is all the values that make $2 (2 u + 1) (u + 1) = 0$ true.

$u = - \frac{1}{2}, - 1$

Step 32

Substitute $\sin (x)$ for $u$ .

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

Step 33

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

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

$\sin (x) = - 1$

Step 34

Solve for $x$ in $\sin (x) = - \frac{1}{2}$ .

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Step 34.1

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

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

Step 34.2

Simplify the right side.

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Step 34.2.1

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

$x = - \frac{π}{6}$

Step 34.3

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.

$x = 2 π + \frac{π}{6} + π$

Step 34.4

Simplify the expression to find the second solution.

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Step 34.4.1

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

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 34.4.2

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

$x = \frac{7 π}{6}$

Step 34.5

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

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Step 34.5.1

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

$\frac{2 π}{| b |}$

Step 34.5.2

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

$\frac{2 π}{| 1 |}$

Step 34.5.3

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

$\frac{2 π}{1}$

Step 34.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 34.6

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

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Step 34.6.1

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

$- \frac{π}{6} + 2 π$

Step 34.6.2

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

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 34.6.3

Combine fractions.

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Step 34.6.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 34.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 34.6.4

Simplify the numerator.

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Step 34.6.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 34.6.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 34.6.5

List the new angles.

$x = \frac{11 π}{6}$

Step 34.7

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 $n$

Step 35

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

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Step 35.1

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

$x = \arcsin (- 1)$

Step 35.2

Simplify the right side.

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Step 35.2.1

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

$x = - \frac{π}{2}$

Step 35.3

$x = 2 π + \frac{π}{2} + π$

Step 35.4

Simplify the expression to find the second solution.

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Step 35.4.1

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

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 35.4.2

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

$x = \frac{3 π}{2}$

Step 35.5

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

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Step 35.5.1

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

$\frac{2 π}{| b |}$

Step 35.5.2

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

$\frac{2 π}{| 1 |}$

Step 35.5.3

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

$\frac{2 π}{1}$

Step 35.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 35.6

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

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Step 35.6.1

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

$- \frac{π}{2} + 2 π$

Step 35.6.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 35.6.3

Combine fractions.

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Step 35.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 35.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 35.6.4

Simplify the numerator.

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Step 35.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 35.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 35.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 35.7

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

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

Step 36

List all of the solutions.

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 37

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$