Solve for x sin(x)*cos(x)=( square root of 3)/4

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

Square both sides of the equation.

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

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

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

Simplify ${(\frac{\sqrt{3}}{4})}^{2}$ .

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Apply the product rule to $\frac{\sqrt{3}}{4}$ .

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

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

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Rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

$\sin^{2} (x) \cos^{2} (x) = \frac{3^{\frac{1}{2} \cdot 2}}{4^{2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

Raise $4$ to the power of $2$ .

$\sin^{2} (x) \cos^{2} (x) = \frac{3}{16}$

Move $\frac{3}{16}$ to the left side of the equation by subtracting it from both sides.

$\sin^{2} (x) \cos^{2} (x) - \frac{3}{16} = 0$

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

$1 - \cos^{2} (x) \cos^{2} (x) - \frac{3}{16} = 0$

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

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

$1 - (\cos^{2} (x) \cos^{2} (x)) - \frac{3}{16} = 0$

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

$1 - {\cos (x)}^{2 + 2} - \frac{3}{16} = 0$

Add $2$ and $2$ .

$1 - \cos^{4} (x) - \frac{3}{16} = 0$

To write $\frac{1}{1}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- \cos^{4} (x) + \frac{1}{1} \cdot \frac{16}{16} - \frac{3}{16} = 0$

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$- \cos^{4} (x) + \frac{1 \cdot 16}{1 \cdot 16} - \frac{3}{16} = 0$

Multiply $16$ by $1$ .

$- \cos^{4} (x) + \frac{1 \cdot 16}{16} - \frac{3}{16} = 0$

Combine the numerators over the common denominator.

$- \cos^{4} (x) + \frac{1 \cdot 16 - 3}{16} = 0$

Simplify the numerator.

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Multiply $1$ by $16$ .

$- \cos^{4} (x) + \frac{16 - 3}{16} = 0$

Subtract $3$ from $16$ .

$- \cos^{4} (x) + \frac{13}{16} = 0$

Subtract $\frac{13}{16}$ from both sides of the equation.

$- \cos^{4} (x) = - \frac{13}{16}$

Multiply each term in $- \cos^{4} (x) = - \frac{13}{16}$ by $- 1$

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Multiply each term in $- \cos^{4} (x) = - \frac{13}{16}$ by $- 1$ .

$(- \cos^{4} (x)) \cdot - 1 = - \frac{13}{16} \cdot - 1$

Multiply $- \cos^{4} (x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 \cos^{4} (x) = - \frac{13}{16} \cdot - 1$

Multiply $\cos^{4} (x)$ by $1$ .

$\cos^{4} (x) = - \frac{13}{16} \cdot - 1$

Multiply $- \frac{13}{16} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$\cos^{4} (x) = 1 (\frac{13}{16})$

Multiply $\frac{13}{16}$ by $1$ .

$\cos^{4} (x) = \frac{13}{16}$

Take the $4th$ root of both sides of the $equation$ to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt[4]{\frac{13}{16}}$

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 $\sqrt[4]{\frac{13}{16}}$ as $\frac{\sqrt[4]{13}}{\sqrt[4]{16}}$ .

$\cos (x) = \pm \frac{\sqrt[4]{13}}{\sqrt[4]{16}}$

Simplify the denominator.

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Rewrite $16$ as $2^{4}$ .

$\cos (x) = \pm \frac{\sqrt[4]{13}}{\sqrt[4]{2^{4}}}$

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

$\cos (x) = \pm \frac{\sqrt[4]{13}}{2}$

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

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

$\cos (x) = \frac{\sqrt[4]{13}}{2}$

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

$\cos (x) = - \frac{\sqrt[4]{13}}{2}$

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

$\cos (x) = \frac{\sqrt[4]{13}}{2}, - \frac{\sqrt[4]{13}}{2}$

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

$\cos (x) = \frac{\sqrt[4]{13}}{2}$

$\cos (x) = - \frac{\sqrt[4]{13}}{2}$

Set up the equation to solve for $x$ .

$\cos (x) = \frac{\sqrt[4]{13}}{2}$

Solve the equation for $x$ .

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

$x = \arccos (\frac{\sqrt[4]{13}}{2})$

Evaluate $\arccos (\frac{\sqrt[4]{13}}{2})$ .

$x = 0.31943033$

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.

$x = 2 (3.14159265) - 0.31943033$

Simplify $2 (3.14159265) - 0.31943033$ .

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Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.31943033$

Subtract $0.31943033$ from $6.2831853$ .

$x = 5.96375497$

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 $1$ .

$2 π$

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

$x = 0.31943033 + 2 π n, 5.96375497 + 2 π n$ , for any integer $n$

Set up the equation to solve for $x$ .

$\cos (x) = - \frac{\sqrt[4]{13}}{2}$

Solve the equation for $x$ .

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

$x = \arccos (- \frac{\sqrt[4]{13}}{2})$

Evaluate $\arccos (- \frac{\sqrt[4]{13}}{2})$ .

$x = 2.82216231$

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.

$x = 2 (3.14159265) - 2.82216231$

Simplify $2 (3.14159265) - 2.82216231$ .

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Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.82216231$

Subtract $2.82216231$ from $6.2831853$ .

$x = 3.46102299$

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 $1$ .

$2 π$

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

$x = 2.82216231 + 2 π n, 3.46102299 + 2 π n$ , for any integer $n$

List all of the results found in the previous steps.

$x = 0.31943033 + 2 π n, 2.82216231 + 2 π n, 3.46102299 + 2 π n, 5.96375497 + 2 π n$ , for any integer $n$

The complete solution is the set of all solutions.

$x = 0.31943033 + 2 π n, 5.96375497 + 2 π n, 2.82216231 + 2 π n, 3.46102299 + 2 π n$ , for any integer $n$

Consolidate the answers.

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Consolidate $0.31943033 + 2 π n$ and $3.46102299 + 2 π n$ to $0.31943033 + π n$ .

$x = 0.31943033 + π n, 5.96375497 + 2 π n, 2.82216231 + 2 π n$ , for any integer $n$

Consolidate $5.96375497 + 2 π n$ and $2.82216231 + 2 π n$ to $2.82216231 + π n$ .

$x = 0.31943033 + π n, 2.82216231 + π n$ , for any integer $n$

Exclude the solutions that do not make $\sin (x) \cdot \cos (x) = \frac{\sqrt{3}}{4}$ true.

No solution

Solve for x sin(x)*cos(x)=( square root of 3)/4

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