Why is cos(π) and cos( -π) both equal to -1?

To understand why both cos(π) and cos(-π) equal -1, we need to delve into the properties of the cosine function and how it behaves on the unit circle. The cosine function is periodic and even, which plays a crucial role in this scenario.

The Unit Circle and Cosine Function

The cosine of an angle in the unit circle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.

Evaluating cos(π)

When we evaluate cos(π), we are looking at the angle π radians. On the unit circle, this angle corresponds to the point (-1, 0). Therefore:

• cos(π) = x-coordinate of the point at angle π = -1

Evaluating cos(-π)

Now, let’s consider cos(-π). The negative sign indicates that we are measuring the angle in the clockwise direction. However, an angle of -π radians is equivalent to moving π radians in the counterclockwise direction, landing us at the same point on the unit circle, which is also (-1, 0). Thus:

• cos(-π) = x-coordinate of the point at angle -π = -1

Properties of the Cosine Function

Another important aspect to note is that the cosine function is an even function. This means that:

• cos(-x) = cos(x) for any angle x

Applying this property, we can see that:

• cos(-π) = cos(π) = -1

Visualizing the Concept

To visualize this, imagine standing at the origin of the unit circle. If you rotate π radians counterclockwise, you point directly to the left at (-1, 0). If you rotate -π radians, you are effectively moving in the opposite direction but still end up at the same point. This symmetry around the y-axis illustrates why both angles yield the same cosine value.

Summary

In summary, both cos(π) and cos(-π) equal -1 because they correspond to the same point on the unit circle, and the cosine function's even property confirms this relationship. Understanding these concepts not only clarifies this specific question but also strengthens your grasp of trigonometric functions and their properties.