Question

Find the exact values of s in the interval [0,2π] that satisfy the given condition.

Answer 1

The exact values of s in the interval
`\([0,2\pi]\)` that satisfy the given condition are
`\(s = (\pi)/(6)\) and \(s = (5\pi)/(6)\).`

Step-by-step explanation:

To find the exact values of s in the interval
`\([0,2\pi]\)`satisfying the given condition, we can set up the equation
`\(\sin(s) = (√(3))/(2)\).` In the unit circle,
`\(\sin(s)\)`represents the y-coordinate of a point on the circle corresponding to the angle s. The value
`\((√(3))/(2)\)` corresponds to the y-coordinate of the point at
`\((\pi)/(3)\) and \((5\pi)/(3)\)` on the unit circle.

Therefore, the solutions to the equation are
`\(s = (\pi)/(3)\) and \(s = (5\pi)/(3)\)`. However, since we are looking for solutions in the interval
`\([0,2\pi]\),`we need to discard the
`\((5\pi)/(3)\)` solution as it falls outside this range. Thus, the exact values of ssatisfying the given condition in the specified interval are
`\(s = (\pi)/(3)\) and \(s = (5\pi)/(3)\).`

In conclusion, understanding the unit circle and the trigonometric values associated with different angles allows us to identify the solutions to trigonometric equations. In this case, the solutions lie in the specified interval, providing the exact values of s that satisfy the given condition.