Exploring the Trigonometric Identity: cos(-x) cos(x)

Many students often confuse the trigonometric identity involving the cosine function and the negative angle. In this article, we will explore and clarify why cos(-x) cos(x). While it is often mistakenly believed that cos(-x) -cos(x), we will demonstrate that this statement is actually incorrect and explain why the correct identity holds.

Understanding Even Functions

The cosine function, cos(x), is a classic example of an even function. By definition, a function f(x) is even if it satisfies the condition:

f(-x) f(x)

Applying this to the cosine function, we get:

cos(-x) cos(x)

This property indicates that the cosine function is symmetric with respect to the y-axis. If you were to reflect any point on the graph of the cosine function over the y-axis, its y-coordinate (which is the cosine value) would remain the same.

The Unit Circle Perspective

The unit circle provides a useful tool for visualizing the behavior of trigonometric functions, including cosine. On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. When we consider an angle -x, its terminal side is a reflection of the angle x across the y-axis. Consequently, the x-coordinate (and thus the cosine value) remains unchanged.

Geometric Representation

Imagine the unit circle centered at the origin of a coordinate system. If we have an angle x, the point where its terminal side intersects the unit circle is (cos(x), sin(x)). Now, if we rotate the angle by -x, the terminal side of the angle will now point in the direction of the angle x, but in the opposite direction. Geometrically, this reflection means that the x-coordinate remains the same (cos(-x) cos(x)), while the y-coordinate is negated (sin(-x) -sin(x)).

Graphical Interpretation

A graphical representation of the cosine function reinforces this concept. If you plot the function y cos(x), you will notice that the graph is symmetric about the y-axis. This symmetry is a visual manifestation of the even property of the cosine function. Hence, for any x, the value of cos(-x) will be the same as cos(x).

Conclusion

In summary, the correct trigonometric identity for the cosine function and a negative angle is:

cos(-x) cos(x)

This identity is a result of the even nature of the cosine function, which is reflected in its symmetry about the y-axis. If you are still unclear about this concept or need further clarification, feel free to ask!