Solve for x 2sin(2x)cos(x)+2cos(2x)sin(x) = square root of 3

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Trigonometry

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

Square both sides of the equation.

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

Simplify terms.

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Simplify each term.

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Apply the sine double-angle identity.

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

Multiply $2$ by $2$ to get $4$ .

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

Remove parentheses around $\sin (x) \cos (x)$ .

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

Simplify $4 \sin (x) \cos (x) \cos (x)$ .

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

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

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

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

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

${(4 \sin (x) {\cos (x)}^{1 + 1} + 2 \cos (2 x) \sin (x))}^{2} = {(\sqrt{3})}^{2}$

Add $1$ and $1$ to get $2$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ to get $- 2$ .

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

Apply the distributive property.

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

Move $\sin (x)$ .

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

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

${(4 \sin (x) \cos^{2} (x) + (2 \cos^{2} (x) \sin (x) - 2 {\sin (x)}^{1 + 2}))}^{2} = {(\sqrt{3})}^{2}$

Add $1$ and $2$ to get $3$ .

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

Simplify by adding terms.

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Remove unnecessary parentheses.

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

Reorder terms.

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

Add $4 \cos^{2} (x) \sin (x)$ and $2 \cos^{2} (x) \sin (x)$ to get $6 \cos^{2} (x) \sin (x)$ .

${(6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x))}^{2} = {(\sqrt{3})}^{2}$

Rewrite ${(6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x))}^{2}$ as $(6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) (6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x))$ .

$(6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) (6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Expand $(6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) (6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x))$ using the FOIL Method.

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Apply the distributive property.

$6 \cos^{2} (x) \sin (x) (6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x) - 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Simplify and combine like terms.

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Simplify each term.

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

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

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

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

Add $2$ and $2$ to get $4$ .

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

Simplify $6 \sin (x) (6 \cos^{4} (x) \sin (x))$ .

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Multiply $6$ by $6$ to get $36$ .

$36 \sin (x) (\cos^{4} (x) \sin (x)) + 6 \cos^{2} (x) \sin (x) (- 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

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

$36 (\sin (x) \sin (x)) \cos^{4} (x) + 6 \cos^{2} (x) \sin (x) (- 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

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

$36 (\sin (x) \sin (x)) \cos^{4} (x) + 6 \cos^{2} (x) \sin (x) (- 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

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

$36 {\sin (x)}^{1 + 1} \cos^{4} (x) + 6 \cos^{2} (x) \sin (x) (- 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Add $1$ and $1$ to get $2$ .

$36 \sin^{2} (x) \cos^{4} (x) + 6 \cos^{2} (x) \sin (x) (- 2 \sin^{3} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Move $\sin (x)$ .

$36 \sin^{2} (x) \cos^{4} (x) + 6 \cos^{2} (x) (- 2 (\sin (x) \sin^{3} (x))) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

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

$36 \sin^{2} (x) \cos^{4} (x) + 6 \cos^{2} (x) (- 2 {\sin (x)}^{1 + 3}) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Add $1$ and $3$ to get $4$ .

$36 \sin^{2} (x) \cos^{4} (x) + 6 \cos^{2} (x) (- 2 \sin^{4} (x)) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Multiply $- 2$ by $6$ to get $- 12$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 \sin^{3} (x) (6 \cos^{2} (x) \sin (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Move $\sin^{3} (x)$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (6 \cos^{2} (x) (\sin^{3} (x) \sin (x))) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

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

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (6 \cos^{2} (x) {\sin (x)}^{3 + 1}) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Add $3$ and $1$ to get $4$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (6 \cos^{2} (x) \sin^{4} (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Multiply $6$ by $- 2$ to get $- 12$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 (\cos^{2} (x) \sin^{4} (x)) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Remove parentheses around $\cos^{2} (x) \sin^{4} (x)$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 \sin^{3} (x) (- 2 \sin^{3} (x)) = {(\sqrt{3})}^{2}$

Move $\sin^{3} (x)$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (- 2 (\sin^{3} (x) \sin^{3} (x))) = {(\sqrt{3})}^{2}$

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

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (- 2 {\sin (x)}^{3 + 3}) = {(\sqrt{3})}^{2}$

Add $3$ and $3$ to get $6$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 2 (- 2 \sin^{6} (x)) = {(\sqrt{3})}^{2}$

Multiply $- 2$ by $- 2$ to get $4$ .

$36 \sin^{2} (x) \cos^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) - 12 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = {(\sqrt{3})}^{2}$

Subtract $12 \cos^{2} (x) \sin^{4} (x)$ from $- 12 \cos^{2} (x) \sin^{4} (x)$ to get $- 24 \cos^{2} (x) \sin^{4} (x)$ .

$36 \sin^{2} (x) \cos^{4} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = {(\sqrt{3})}^{2}$

Remove parentheses around $\sqrt{3}$ .

$36 \sin^{2} (x) \cos^{4} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = {\sqrt{3}}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

$36 \sin^{2} (x) \cos^{4} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 3$

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

$1 - \cos^{2} (x) \cos^{4} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 3$

Simplify each term.

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Move $\cos^{4} (x)$ .

$1 - (\cos^{4} (x) \cos^{2} (x)) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 3$

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

$1 - {\cos (x)}^{4 + 2} - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 3$

Add $4$ and $2$ to get $6$ .

$1 - \cos^{6} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 3$

Reorder the polynomial $1 - \cos^{6} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x)$ alphabetically from left to right, starting with the highest order term.

$- \cos^{6} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) + 1 = 3$

Move all terms not containing $x$ to the right side of the equation.

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Since $1$ does not contain the variable to solve for, move it to the right side of the equation by subtracting $1$ from both sides.

$- \cos^{6} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = - 1 + 3$

Add $- 1$ and $3$ to get $2$ .

$- \cos^{6} (x) - 24 \cos^{2} (x) \sin^{4} (x) + 4 \sin^{6} (x) = 2$

Substitute $u = \cos^{2} (x)$ into the equation. This will make the quadratic formula easy to use.

$36 \sin^{2} (x) u^{2} - 24 \sin^{4} (x) u + 4 \sin^{6} (x) = 3$

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

Since $4 \sin^{6} (x)$ does not contain the variable to solve for, move it to the right side of the equation by subtracting $4 \sin^{6} (x)$ from both sides.

$36 \sin^{2} (x) u^{2} - 24 \sin^{4} (x) u = - 4 \sin^{6} (x) + 3$

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

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Move $4 \sin^{6} (x)$ to the left side of the equation by adding it to both sides.

$36 \sin^{2} (x) u^{2} - 24 \sin^{4} (x) u + 4 \sin^{6} (x) = 3$

Move $3$ to the left side of the equation by subtracting it from both sides.

$36 \sin^{2} (x) u^{2} - 24 \sin^{4} (x) u + 4 \sin^{6} (x) - 3 = 0$

Substitute the real value of $u = \cos^{2} (x)$ back into the solved equation.

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

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

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

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

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

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

$\cos (x) = \pm 0$

$\pm 0$ is equal to $0$ .

$\cos (x) = 0$

Simplify the expression to find the first solution.

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Take the inverse $c o s i n e$ of both sides of the equation to extract $x$ from inside the $c o s i n e$ .

$x = \arccos (0)$

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

$x = \frac{π}{2}$

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.

$x = 2 π - \frac{π}{2}$

Simplify the expression to find the second solution.

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{2 π}{1} \cdot \frac{2}{2} - \frac{π}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{2 π \cdot 2}{1 \cdot 2} - \frac{π}{2}$

Multiply $2$ by $1$ to get $2$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Factor $π$ out of $2 π \cdot 2 - π$ .

$x = \frac{π (2 \cdot 2 - 1)}{2}$

Multiply $2$ by $2$ to get $4$ .

$x = \frac{π (4 - 1)}{2}$

Subtract $1$ from $4$ to get $3$ .

$x = \frac{π \cdot 3}{2}$

Simplify the expression.

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Move $3$ to the left of the expression $π \cdot 3$ .

$x = \frac{3 \cdot π}{2}$

Multiply $3$ by $π$ to get $3 π$ .

$x = \frac{3 π}{2}$

Find the period.

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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.

$\frac{2 π}{| 1 |}$

Solve the equation.

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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 $1$ to get $2 π$ .

$2 π$

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

$x = \frac{π}{2} \pm 2 π n, \frac{3 π}{2} \pm 2 π n$

Consolidate the answers.

$x = \frac{π}{2} \pm 2 π n$

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