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Find all values of x
Sin2x+sqrt3/2 = 0

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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:

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