The Reason Behind the Equality of Cosine and Its Negation

The equation ( cos(theta) cos(-theta) ) is a fascinating result arising from the even nature of the cosine function. This property is deeply rooted in the foundational concepts of trigonometry and can be explained and demonstrated through a variety of methods, including Euler's formula, the unit circle, and the symmetry of the cosine function.

Definition of Cosine and the Unit Circle

The cosine function is defined in terms of the unit circle. For any angle ( theta ), the cosine of that angle corresponds to the x-coordinate of the point on the unit circle at that angle. This definition is crucial to understanding why ( cos(theta) cos(-theta) ).

Symmetry of the Unit Circle

The unit circle exhibits symmetry about the y-axis. This means that for any angle ( theta ), the point corresponding to ( -theta ) (the angle measured in the opposite direction) has the same x-coordinate as the point corresponding to ( theta ).

Mathematical Expression of Symmetry

Mathematically, this symmetry can be expressed as:

( cos(-theta) x ) and ( cos(theta) x )

Since both angles ( theta ) and ( -theta ) yield the same x-coordinate, we conclude that:

( cos(theta) cos(-theta) )

Even Function

Functions that satisfy the property ( f(-x) f(x) ) for all ( x ) are called even functions. Since ( cos(theta) ) meets this criterion, it is classified as an even function.

Exploring Cosine Through Euler's Formula

Euler's formula provides another compelling way to understand why ( cos(theta) cos(-theta) ). Euler's formula states that:

[ e^{itheta} cos(theta) isin(theta) ]

Applying this to ( -theta ) gives:

[ e^{i(-theta)} cos(-theta) isin(-theta) ]

The conjugate of a complex number is obtained by replacing ( i ) with ( -i ). The conjugate of Euler's formula for ( theta ) is:

[ e^{-itheta} cos(theta) - isin(theta) ]

Since the left-hand sides of the original and conjugate equations are equal, their right-hand sides must also be equal. Equating the real and imaginary parts, we get:

( cos(-theta) cos(theta) )

( sin(-theta) -sin(theta) )

Taylor Series Perspective

The Taylor series of ( cos(theta) ) is the same as the series for ( cos(-theta) ) because it contains only even powers:

[ cos(theta) 1 - frac{theta^2}{2!} frac{theta^4}{4!} - cdots ]

[ cos(-theta) 1 - frac{(-theta)^2}{2!} frac{(-theta)^4}{4!} - cdots 1 - frac{theta^2}{2!} frac{theta^4}{4!} - cdots cos(theta) ]

Similarly, using the same reasoning ( text{mutatis mutandis} ), we can prove that ( sin(theta) -sin(-theta) ).

Conclusion

In summation, the equality ( cos(theta) cos(-theta) ) is due to the even nature of the cosine function, the symmetry of the unit circle, and the application of Euler's formula. Understanding these relationships deepens our appreciation for the elegant patterns and symmetries inherent in trigonometry.