Final answer:
The sine function has an amplitude of 1, a period of 2π radians, a domain of all real numbers, a range of [-1, 1], and x-intercepts at multiples of π. These properties relate to the unit circle where the sine corresponds to y-coordinates.
Step-by-step explanation:
Properties of the Sine Function
When analyzing the graph of the sine function, f(theta) = sin(theta), several key properties are observed.
- Amplitude is the maximum displacement from the midpoint, which in this case, is 1 (since sine oscillates between +1 and -1).
- Period is the length of one complete cycle of the graph. For sine, this is 2π radians (360 degrees).
- Domain of the sine function is all real numbers, meaning θ can take any value from negative to positive infinity.
- Range of sine is between -1 and +1, inclusive, representing the maximum and minimum values sine can achieve.
- x-intercepts of the sine wave occur at multiples of π, where the sine of angle θ is zero.
These properties are easily understood when looking at the unit circle, where the sine of an angle is the y-coordinate of a point on the circle's circumference. The periodic nature of the sine function is reflected in the circular path that repeats every 360 degrees or 2π radians.