Answer:
To find all values of x that satisfy the equation Sin(2x) + sqrt(3)/2 = 0, we can consider the following steps:
1. Understand the given equation:
The equation Sin(2x) + sqrt(3)/2 = 0 represents a trigonometric equation involving the sine function. We want to find the values of x that make this equation true.
2. Use the properties of the sine function:
The sine function has a period of 2π. This means that the values of x that satisfy the equation will repeat every 2π units.
3. Solve the equation:
Subtract sqrt(3)/2 from both sides of the equation:
Sin(2x) = -sqrt(3)/2
4. Determine the reference angle:
We know that Sin(π/3) = sqrt(3)/2. Since the sine function is negative in the third and fourth quadrants, we can write the equation as:
2x = π - π/3
or
2x = 2π - π/3
5. Solve for x:
Divide both sides of the equation by 2:
x = (π - π/3)/2
or
x = (2π - π/3)/2
6. Simplify the expressions:
For the first equation:
x = π/2 - π/6
Simplifying further:
x = π/3
For the second equation:
x = 2π/2 - π/6
Simplifying further:
x = 5π/6
7. Find the general solution:
Since the sine function has a period of 2π, we can add multiples of 2π to the solutions we found in step 6 to find all the values of x that satisfy the equation.
The general solution is:
x = π/3 + 2πk
x = 5π/6 + 2πk
where k is an integer.
Therefore, the values of x that satisfy the equation Sin(2x) + sqrt(3)/2 = 0 are x = π/3 + 2πk and x = 5π/6 + 2πk, where k is an integer.
Explanation: