Short Answer 
The unit circle, centered at the origin with a radius of 1, is crucial for understanding trigonometric functions, particularly the sine function. When sketching the sine function, plot angles from 0 to 2œA along the x-axis and mark corresponding y-values, showing the patterns as the terminal ray rotates. After a full rotation, the x-coordinate remains constant at 1, illustrating the horizontal distance on the unit circle.
Step 1: Understand the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin (0,0) of the xy-plane. It’s essential in trigonometry as it helps visualize angles and their corresponding coordinates. When observing the rotation of a terminal ray, remember that it rotates counterclockwise from the positive x-axis and represents angles in radians.
Step 2: Sketch the Coordinates
To sketch the graph of the sine function using the unit circle, label your axes. Ensure to include angles ranging from 0 to 2‚àöe¬¨A on the x-axis and values from -1 to 1 on the y-axis. As you plot the terminal ray’s movement:
- From 0 to √e¬A/2, the ray moves upward, increasing the y-coordinate from 0 to 1.
- From √e¬A/2 to √e¬A, the ray descends, decreasing the y-coordinate back to 0.
- Continue this pattern for the third and fourth quadrants, completing a full wave pattern.
Step 3: Determine the X-coordinate
After a full rotation of 2√e¬A radians, the terminal ray returns to the same point as when √e¬∏ = 0, which is on the positive x-axis. Since this is a full circle, the x-coordinate at this position is consistently 1, representing the horizontal distance of point P from the origin on the unit circle.