Solve for x sin(x)=( square root of 3)/2 , cos(x)=-1/2

$\sin (x) = \frac{\sqrt{3}}{2}$ , $\cos (x) = - \frac{1}{2}$

Step 1

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

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

Step 2

Simplify the right side.

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

$x = \frac{π}{3}$

Step 3

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.

$x = π - \frac{π}{3}$

Step 4

Simplify $π - \frac{π}{3}$ .

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

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

Combine fractions.

Combine $π$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π}{3}$

Simplify the numerator.

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π}{3}$

Subtract $π$ from $3 π$ .

$x = \frac{2 π}{3}$

Step 5

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

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.

$\frac{2 π}{| 1 |}$

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 $1$ .

$2 π$

Step 6

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

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

Step 7

Solve the equation for $π$ .

Rewrite the equation as $\frac{π}{3} + 2 π n = x$ .

$\frac{π}{3} + 2 π n = x$

$\frac{2 π}{3} + 2 π n$

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

Subtract $\frac{π}{3}$ from both sides of the equation.

$2 π n = x - \frac{π}{3}$

$\frac{2 π}{3} + 2 π n$

Subtract $2 π n$ from both sides of the equation.

$0 = x - \frac{π}{3} - 2 π n$

$\frac{2 π}{3} + 2 π n$

$0 = x - \frac{π}{3} - 2 π n$

$\frac{2 π}{3} + 2 π n$

Rewrite the equation as $x - \frac{π}{3} - 2 π n = 0$ .

$x - \frac{π}{3} - 2 π n = 0$

$\frac{2 π}{3} + 2 π n$

$x - \frac{π}{3} - 2 π n = 0$

$\frac{2 π}{3} + 2 π n$

Step 8

The simplified system is the arbitrary solution of the original system of equations.

$x - \frac{π}{3} - 2 π n = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Step 9

Reduce the system.

$x - \frac{π}{3} - 2 π (\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Step 10

Simplify the left side.

Simplify $x - \frac{π}{3} - 2 π (\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n))$ .

Simplify each term.

Multiply $2$ by $0$ .

$x - \frac{π}{3} - 2 π (\frac{0}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Combine the opposite terms in $\frac{0}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$ .

Divide $0$ by $3$ .

$x - \frac{π}{3} - 2 π (0 + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Add $0$ and $2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$ .

$x - \frac{π}{3} - 2 π (2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} - 2 π (2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $2$ by $0$ .

$x - \frac{π}{3} - 2 π (0 \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $2$ by $0$ .

$x - \frac{π}{3} - 2 π (0 \cdot (\frac{0}{3} + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Combine the opposite terms in $\frac{0}{3} + 2 (0) \cdot n$ .

Divide $0$ by $3$ .

$x - \frac{π}{3} - 2 π (0 \cdot (0 + 2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Add $0$ and $2 (0) \cdot n$ .

$x - \frac{π}{3} - 2 π (0 \cdot (2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} - 2 π (0 \cdot (2 (0) \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $2$ by $0$ .

$x - \frac{π}{3} - 2 π (0 \cdot (0 \cdot n)) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $0$ by $n$ .

$x - \frac{π}{3} - 2 π (0 \cdot 0) = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $0$ by $0$ .

$x - \frac{π}{3} - 2 π \cdot 0 = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $- 2 π \cdot 0$ .

Multiply $0$ by $- 2$ .

$x - \frac{π}{3} + 0 π = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Multiply $0$ by $π$ .

$x - \frac{π}{3} + 0 = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} + 0 = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} + 0 = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Add $x - \frac{π}{3}$ and $0$ .

$x - \frac{π}{3} = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

$x - \frac{π}{3} = 0$

$\frac{2 (0)}{3} + 2 (0) \cdot (\frac{2 (0)}{3} + 2 (0) \cdot n)$

Solve for x sin(x)=( square root of 3)/2 , cos(x)=-1/2

Related Questions

Support LCMGCF.com by disabling your adblocker.