Determine how many integer solutions the inequality has on the interval (0;2\pi ) \bf\\sin\Big(2x+(\pi )/(3) \Big)\leq (1)/(2)

Question

Determine how many integer solutions the inequality has on the interval
`(0;2\pi )`


`\bf\\sin\Big(2x+(\pi )/(3) \Big)\leq (1)/(2)`

Answer 1

• There are 5 integer solutions: x = {1, 2, 4, 5, 6}

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Graph the function y = sin(2x + π/3), take the section of it within interval (0, 2π).

Consider integer values of x in the same interval:

• x ∈ (0, 2π ) or x = {1, 2, 3, 4, 5, 6}

and see at what integer values of x the function doesn't exceed the value of 1/2.

See attached.

As we see x = {1, 2, 4, 5, 6} is the set of integer solutions.

Another solution is to plug in all integer values 1 to 6 into inequality and evaluate.