Yes, sin(0) is indeed equal to cos(90), and here’s the detailed reasoning behind this conclusion.
In trigonometry, the sine and cosine functions are defined based on the unit circle, which is a circle with a radius of 1 centered at the origin of a Cartesian coordinate system. The angles are typically measured in degrees or radians, and they correspond to points on this circle.
1. Understanding sin(0): When we evaluate sin(0), we are looking at the sine of an angle measuring 0 degrees. On the unit circle, this angle corresponds to the point (1, 0). The sine of an angle is defined as the y-coordinate of the corresponding point on the unit circle. Therefore:
- sin(0) = y-coordinate of (1, 0) = 0
2. Understanding cos(90): Next, we evaluate cos(90), which is the cosine of an angle measuring 90 degrees. On the unit circle, this angle corresponds to the point (0, 1). The cosine of an angle is defined as the x-coordinate of the corresponding point on the unit circle. Therefore:
- cos(90) = x-coordinate of (0, 1) = 0
3. Conclusion: Since both sin(0) and cos(90) yield the value 0, we can conclude that:
sin(0) = cos(90)
This relationship is part of the fundamental identities in trigonometry and helps illustrate the interconnected nature of these functions. Understanding these relationships not only deepens our comprehension of trigonometry but also highlights the beauty of mathematics as a coherent system.