Why is the Range of sin x Between -1 and 1?
When discussing trigonometric functions, the sine function is one of the most fundamental and widely used. Specifically, the range of sin x is between -1 and 1. In this article, we will delve into why this is the case and explore the underlying geometry of the unit circle.
The Unit Circle and the Sine Function
The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. It is a crucial tool in trigonometry because it provides a geometric interpretation of trigonometric functions. The sine of an angle in standard position is defined as the y-coordinate of the point where the terminal ray of the angle intersects the unit circle.
Consider an angle in standard position where the initial ray is the positive x-axis. As the angle changes, the terminal ray rotates around the origin, and the point of intersection of this terminal ray with the unit circle changes. This point has coordinates (x, y), where x is the cosine of the angle and y is the sine of the angle.
The Maximum and Minimum Values
The key insight is in understanding the maximum and minimum y-coordinates on the unit circle. Since the radius of the unit circle is 1, the circle extends from -1 to 1 along both the x and y axes. Consequently, the y-coordinates can never exceed 1 or fall below -1. This is because the y-coordinate of any point on the unit circle is the vertical component of that point relative to the origin, and the maximum and minimum vertical distances are exactly 1 unit.
Mathematically, the maximum value of sin x is 1, which occurs when the terminal ray lies on the positive y-axis. This corresponds to an angle of 90 degrees (or π/2 radians) in standard position. Conversely, the minimum value of sin x is -1, which occurs when the terminal ray lies on the negative y-axis, corresponding to an angle of 270 degrees (or 3π/2 radians).
The Continuity of the Sine Function
The sine function is continuous, meaning there are no breaks or jumps in its graph. This continuity is a consequence of the smooth rotation of the terminal ray around the origin. As a result, for every value of x, there is a corresponding value of sin x between -1 and 1.
To put it succinctly, because the maximum value of sin x is 1 and the minimum value is -1, and the function is continuous, it follows that sin x will take on all values between -1 and 1 for all real numbers x. This is a direct result of the properties of the unit circle and the definition of the sine function.
Conclusion
The range of the sine function, sin x, is between -1 and 1 due to the geometric constraints of the unit circle. Understanding this range helps in various applications, from calculus to physics, where trigonometric functions are essential. By comprehending the geometry of the unit circle and the properties of continuous functions, one can gain a deeper appreciation for this fundamental relationship in mathematics.
Frequently Asked Questions
Q: Can the sine function take on values outside the range -1 to 1?
No, the sine function is defined such that its range is between -1 and 1. The unit circle ensures that the y-coordinate (sin x) cannot exceed 1 or go below -1.
Q: Is the sine function periodic?
Yes, the sine function is periodic with a period of 2π. This means that sin(x 2π) sin(x) for all real numbers x.
Q: How does the range -1 to 1 affect the graph of the sine function?
The range of -1 to 1 bounds the amplitude of the sine function's graph, making it oscillate between -1 and 1 for all x-values.