The value of sin 3π is 0. Here's how to determine that:
Understanding Sine and the Unit Circle
The sine function, denoted as sin(θ), relates an angle θ to the y-coordinate of a point on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. When evaluating trigonometric functions of angles like 3π, we are essentially finding the coordinates of the point where the terminal side of the angle intersects the unit circle.
Visualizing 3π Radians
- A full revolution around the unit circle is equal to 2π radians.
- Therefore, 3π radians is equivalent to one and a half revolutions (2π + π).
- Starting from the positive x-axis (0 radians), rotating 2π radians brings you back to the starting point.
- Then, rotating an additional π radians (180 degrees) places you on the negative x-axis.
Coordinates on the Unit Circle
The point on the unit circle corresponding to an angle of 3π is (-1, 0).
Determining sin 3π
Since sine corresponds to the y-coordinate of the point on the unit circle, and the y-coordinate of the point (-1, 0) is 0, then:
sin 3π = 0
In summary: Evaluating sin 3π involves understanding its relationship to the unit circle, visualizing the angle's position on the circle, and identifying the corresponding y-coordinate.
