Trigonometry Examples

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

Step 1

Subtract $3$ from both sides of the equation.

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

Step 2

Square both sides of the equation.

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

Step 3

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

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Step 3.1

Apply the product rule to $3 \sin (x)$ .

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

Step 3.2

Raise $3$ to the power of $2$ .

$9 \sin^{2} (x) = {(2 \cos^{2} (x) - 3)}^{2}$

Step 4

Simplify ${(2 \cos^{2} (x) - 3)}^{2}$ .

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Step 4.1

Rewrite ${(2 \cos^{2} (x) - 3)}^{2}$ as $(2 \cos^{2} (x) - 3) (2 \cos^{2} (x) - 3)$ .

$9 \sin^{2} (x) = (2 \cos^{2} (x) - 3) (2 \cos^{2} (x) - 3)$

Step 4.2

Expand $(2 \cos^{2} (x) - 3) (2 \cos^{2} (x) - 3)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$9 \sin^{2} (x) = 2 \cos^{2} (x) (2 \cos^{2} (x) - 3) - 3 (2 \cos^{2} (x) - 3)$

Step 4.2.2

Apply the distributive property.

$9 \sin^{2} (x) = 2 \cos^{2} (x) (2 \cos^{2} (x)) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x) - 3)$

Step 4.2.3

Apply the distributive property.

$9 \sin^{2} (x) = 2 \cos^{2} (x) (2 \cos^{2} (x)) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$9 \sin^{2} (x) = 2 \cdot 2 \cos^{2} (x) \cos^{2} (x) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Move $\cos^{2} (x)$ .

$9 \sin^{2} (x) = 2 \cdot 2 (\cos^{2} (x) \cos^{2} (x)) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.2.2

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

$9 \sin^{2} (x) = 2 \cdot 2 {\cos (x)}^{2 + 2} + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.2.3

Add $2$ and $2$ .

$9 \sin^{2} (x) = 2 \cdot 2 \cos^{4} (x) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.3

Multiply $2$ by $2$ .

$9 \sin^{2} (x) = 4 \cos^{4} (x) + 2 \cos^{2} (x) \cdot - 3 - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.4

Multiply $- 3$ by $2$ .

$9 \sin^{2} (x) = 4 \cos^{4} (x) - 6 \cos^{2} (x) - 3 (2 \cos^{2} (x)) - 3 \cdot - 3$

Step 4.3.1.5

Multiply $2$ by $- 3$ .

$9 \sin^{2} (x) = 4 \cos^{4} (x) - 6 \cos^{2} (x) - 6 \cos^{2} (x) - 3 \cdot - 3$

Step 4.3.1.6

Multiply $- 3$ by $- 3$ .

$9 \sin^{2} (x) = 4 \cos^{4} (x) - 6 \cos^{2} (x) - 6 \cos^{2} (x) + 9$

Step 4.3.2

Subtract $6 \cos^{2} (x)$ from $- 6 \cos^{2} (x)$ .

$9 \sin^{2} (x) = 4 \cos^{4} (x) - 12 \cos^{2} (x) + 9$

Step 5

Move all the expressions to the left side of the equation.

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Step 5.1

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

$9 \sin^{2} (x) - 4 \cos^{4} (x) = - 12 \cos^{2} (x) + 9$

Step 5.2

Add $12 \cos^{2} (x)$ to both sides of the equation.

$9 \sin^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 9$

Step 5.3

Subtract $9$ from both sides of the equation.

$9 \sin^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) - 9 = 0$

Step 6

Simplify $9 \sin^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) - 9$ .

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Step 6.1

Move $- 9$ .

$9 \sin^{2} (x) - 9 - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.2

Reorder $9 \sin^{2} (x)$ and $- 9$ .

$- 9 + 9 \sin^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.3

Factor $- 9$ out of $- 9$ .

$- 9 (1) + 9 \sin^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.4

Factor $- 9$ out of $9 \sin^{2} (x)$ .

$- 9 (1) - 9 (- \sin^{2} (x)) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.5

Factor $- 9$ out of $- 9 (1) - 9 (- \sin^{2} (x))$ .

$- 9 (1 - \sin^{2} (x)) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.6

Apply pythagorean identity.

$- 9 \cos^{2} (x) - 4 \cos^{4} (x) + 12 \cos^{2} (x) = 0$

Step 6.7

Add $- 9 \cos^{2} (x)$ and $12 \cos^{2} (x)$ .

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

Step 7

Solve for $x$ .

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Step 7.1

Factor the left side of the equation.

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Step 7.1.1

Rewrite $\cos^{4} (x)$ as ${(\cos^{2} (x))}^{2}$ .

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

Step 7.1.2

Let $u = \cos^{2} (x)$ . Substitute $u$ for all occurrences of $\cos^{2} (x)$ .

$3 u - 4 u^{2} = 0$

Step 7.1.3

Factor $u$ out of $3 u - 4 u^{2}$ .

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Step 7.1.3.1

Factor $u$ out of $3 u$ .

$u \cdot 3 - 4 u^{2} = 0$

Step 7.1.3.2

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

$u \cdot 3 + u (- 4 u) = 0$

Step 7.1.3.3

Factor $u$ out of $u \cdot 3 + u (- 4 u)$ .

$u (3 - 4 u) = 0$

Step 7.1.4

Replace all occurrences of $u$ with $\cos^{2} (x)$ .

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

Step 7.2

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

$\cos^{2} (x) = 0$

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

Step 7.3

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

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Step 7.3.1

Set $\cos^{2} (x)$ equal to $0$ .

$\cos^{2} (x) = 0$

Step 7.3.2

Solve $\cos^{2} (x) = 0$ for $x$ .

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Step 7.3.2.1

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

$\cos (x) = \pm \sqrt{0}$

Step 7.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$\cos (x) = \pm \sqrt{0^{2}}$

Step 7.3.2.2.2

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

$\cos (x) = \pm 0$

Step 7.3.2.2.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 7.3.2.3

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

$x = \arccos (0)$

Step 7.3.2.4

Simplify the right side.

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Step 7.3.2.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 7.3.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 7.3.2.6

Simplify $2 π - \frac{π}{2}$ .

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Step 7.3.2.6.1

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

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 7.3.2.6.2

Combine fractions.

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Step 7.3.2.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 7.3.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 7.3.2.6.3

Simplify the numerator.

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Step 7.3.2.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 7.3.2.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 7.3.2.7

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

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Step 7.3.2.7.1

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

$\frac{2 π}{| b |}$

Step 7.3.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3.2.7.3

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

$\frac{2 π}{1}$

Step 7.3.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.3.2.8

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

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

Step 7.4

Set $3 - 4 \cos^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 7.4.1

Set $3 - 4 \cos^{2} (x)$ equal to $0$ .

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

Step 7.4.2

Solve $3 - 4 \cos^{2} (x) = 0$ for $x$ .

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Step 7.4.2.1

Subtract $3$ from both sides of the equation.

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

Step 7.4.2.2

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

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Step 7.4.2.2.1

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

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

Step 7.4.2.2.2

Simplify the left side.

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Step 7.4.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 7.4.2.2.2.1.1

Cancel the common factor.

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

Step 7.4.2.2.2.1.2

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

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

Step 7.4.2.2.3

Simplify the right side.

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Step 7.4.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 7.4.2.3

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

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

Step 7.4.2.4

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

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Step 7.4.2.4.1

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

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

Step 7.4.2.4.2

Simplify the denominator.

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Step 7.4.2.4.2.1

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

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

Step 7.4.2.4.2.2

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

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

Step 7.4.2.5

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

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Step 7.4.2.5.1

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

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

Step 7.4.2.5.2

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

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

Step 7.4.2.5.3

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

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

Step 7.4.2.6

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

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

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

Step 7.4.2.7

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

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Step 7.4.2.7.1

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

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

Step 7.4.2.7.2

Simplify the right side.

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Step 7.4.2.7.2.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 7.4.2.7.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{6}$

Step 7.4.2.7.4

Simplify $2 π - \frac{π}{6}$ .

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Step 7.4.2.7.4.1

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

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

Step 7.4.2.7.4.2

Combine fractions.

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Step 7.4.2.7.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 7.4.2.7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 7.4.2.7.4.3

Simplify the numerator.

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Step 7.4.2.7.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 7.4.2.7.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 7.4.2.7.5

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

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Step 7.4.2.7.5.1

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

$\frac{2 π}{| b |}$

Step 7.4.2.7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.4.2.7.5.3

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

$\frac{2 π}{1}$

Step 7.4.2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.4.2.7.6

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

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

Step 7.4.2.8

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

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Step 7.4.2.8.1

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

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

Step 7.4.2.8.2

Simplify the right side.

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Step 7.4.2.8.2.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 7.4.2.8.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{5 π}{6}$

Step 7.4.2.8.4

Simplify $2 π - \frac{5 π}{6}$ .

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Step 7.4.2.8.4.1

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

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 7.4.2.8.4.2

Combine fractions.

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Step 7.4.2.8.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 7.4.2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 7.4.2.8.4.3

Simplify the numerator.

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Step 7.4.2.8.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 7.4.2.8.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 7.4.2.8.5

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

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Step 7.4.2.8.5.1

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

$\frac{2 π}{| b |}$

Step 7.4.2.8.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.4.2.8.5.3

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

$\frac{2 π}{1}$

Step 7.4.2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.4.2.8.6

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

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

Step 7.4.2.9

List all of the solutions.

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

Step 7.4.2.10

Consolidate the solutions.

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Step 7.4.2.10.1

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

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

Step 7.4.2.10.2

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

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

Step 7.5

The final solution is all the values that make $\cos^{2} (x) (3 - 4 \cos^{2} (x)) = 0$ true.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Step 8

Consolidate the answers.

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Step 8.1

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

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

Step 8.2

Consolidate the answers.

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

Step 9

Verify each of the solutions by substituting them into $3 \sin (x) + 3 = 2 \cos^{2} (x)$ and solving.

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