Final answer:
The exact value of sin(4π/3) is -√3/2. This value is determined by understanding the unit circle and the properties of the sine function in the third quadrant, where 4π/3 radians falls.
Step-by-step explanation:
To determine the exact value of sin(4π/3), we need to understand the unit circle and the properties of sine in various quadrants. The angle 4π/3 radians corresponds to an angle in the third quadrant where sine values are negative. A full circle is 2π radians, so 4π/3 is past the halfway point of the circle (which would be π radians).
Within the unit circle, an angle of π radians represents the point (-1, 0), and as we continue to 4π/3 radians, the corresponding point's y-coordinate (which gives the sine value) will be negative. What we are looking for is the sine of 120 degrees (or π/3 radians), except it will be negative because 4π/3 is in the third quadrant. The value of sin(π/3) is √3/2, which we obtain from the well-known half-angle values. Therefore, the value of sin(4π/3) is -√3/2.
So, breaking it down:
- Identify the quadrant where 4π/3 is located: The third quadrant.
- Determine the reference angle for 4π/3 in the unit circle: π/3 radians (or 60 degrees).
- Note that in the third quadrant, the sine is negative.
- Recognize that sin(π/3) = √3/2.
- Therefore, sin(4π/3) = -√3/2.
This exact value can now be used in calculations involving trigonometric functions.