Find All Complex Solutions sin(x)cos(x)=( square root of 3)/4

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

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $4$ .

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

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $4$ .

$\sin (x) \cos (x) \cdot 4$

Move $4$ to the left of $\sin (x) \cos (x)$ .

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

Multiply each term in $\sin (x) \cos (x) \cdot \frac{4}{4} = \frac{\sqrt{3}}{4}$ by $2$ to eliminate the fractions.

Multiply each term in $\sin (x) \cos (x) \cdot \frac{4}{4} = \frac{\sqrt{3}}{4}$ by $2$ .

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

Simplify the left side.

Simplify the expression.

Reorder $\sin (x) \cos (x) \cdot \frac{4}{4}$ and $2$ .

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

Move parentheses.

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

Apply the sine double-angle identity.

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

Simplify the expression.

Divide $4$ by $4$ .

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

Multiply $\sin (2 x)$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the right side.

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

$2 x = \frac{π}{3}$

Divide each term in $2 x = \frac{π}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{3}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{2}$

Multiply $\frac{π}{3} \cdot \frac{1}{2}$ .

Multiply $\frac{π}{3}$ by $\frac{1}{2}$ .

$x = \frac{π}{3 \cdot 2}$

Multiply $3$ by $2$ .

$x = \frac{π}{6}$

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.

$2 x = π - \frac{π}{3}$

Solve for $x$ .

Simplify.

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 x = π \cdot \frac{3}{3} - \frac{π}{3}$

Combine $π$ and $\frac{3}{3}$ .

$2 x = \frac{π \cdot 3}{3} - \frac{π}{3}$

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 3 - π}{3}$

Subtract $π$ from $π \cdot 3$ .

Reorder $π$ and $3$ .

$2 x = \frac{3 \cdot π - π}{3}$

Subtract $π$ from $3 \cdot π$ .

$2 x = \frac{2 \cdot π}{3}$

Divide each term in $2 x = \frac{2 \cdot π}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{2 \cdot π}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{2 \cdot π}{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{2 \cdot π}{3}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{2 \cdot π}{3}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{2 \cdot π}{3} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 \cdot π$ .

$x = \frac{2 (π)}{3} \cdot \frac{1}{2}$

Cancel the common factor.

$x = \frac{2 π}{3} \cdot \frac{1}{2}$

Rewrite the expression.

$x = \frac{π}{3}$

Find the period of $\sin (2 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

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

Find All Complex Solutions sin(x)cos(x)=( square root of 3)/4

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