Why is sin(0) Equal to 0: A Comprehensive Explanation

The sine function, denoted as sin, is a fundamental trigonometric function that plays a critical role in mathematics and has applications in various fields such as physics, engineering, and computer science. One of the key values of the sine function is sin(0) 0, and this value can be understood through several perspectives.

Unit Circle Definition

On the unit circle, a circle with radius 1 centered at the origin of a coordinate system, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. At an angle of 0 radians or 0circ, the point on the unit circle is (1, 0). The y-coordinate is 0, thus sin(0) 0.

Right Triangle Definition

In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. At 0 degrees, the angle approaches a situation where the length of the opposite side becomes 0, since it effectively collapses to a line while the hypotenuse remains non-zero. Thus, sin(0) 0/hypotenuse 0.

Taylor Series Expansion

The sine function can be expressed as a power series:

sin(x) x - ( frac{x^3}{3!} frac{x^5}{5!} - frac{x^7}{7!} cdots )

When x 0, every term containing x becomes 0, leading to sin(0) 0.

Graphical Interpretation

The graph of the sine function oscillates between -1 and 1, crossing the x-axis at 0. This visually confirms that sin(0) 0. The sine function's value at 0 is 0, and as the angle increases, the value oscillates symmetrically around the x-axis.

Understanding Sine in a Unit Circle

For a unit circle with radius 1 unit, the sine function measures the vertical component of the circle. When the angle is 0 degrees, the vertical component is 0, and the horizontal component is 1.

In this image:

Blue Component: This represents the vertical component of the circle. Value at 0 Degrees: The vertical component is 0 when the angle is 0 degrees, confirming that sin(0) 0.

Additional Examples

To further illustrate, consider a unit circle with the angle at 90 degrees:

Vertical Component: The measure of the vertical component is 1. Horizontal Component: The value of the horizontal component is 0. Since the sine function measures the vertical component, we can conclude that sin(90) 1.

For the angle at 30 degrees:

Vertical Measure: The vertical measure of the line inclined at 30 degrees is 0.5 units. Value of Sine: Thus, sin(30) 0.5.

In summary, sin(0) 0 is a fundamental result that arises from the definitions and properties of the sine function, particularly in relation to the unit circle, right triangles, and Taylor series.