Exploring the Identity for Opposite Cosine (-a) Using the Definition of Cosine Function and Even-Odd Functions

Identities for Oppositescos(-a) =

The identity for Opposite cosine (-a) can be derived using the definition of cosine function and the concept of even and odd functions

The identity for Opposite cosine (-a) can be derived using the definition of cosine function and the concept of even and odd functions.

The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In terms of the coordinate system, it represents the x-coordinate of a point on the unit circle.

Opposite cosine (-a) can be represented as cos(-a). Let’s consider the unit circle in the coordinate system. The angle -a can be obtained by rotating the positive x-axis clockwise by angle a. When we move in the clockwise direction, the y-coordinate becomes negative, and the x-coordinate remains the same.

So, the coordinate on the unit circle can be written as (-cos(a), -sin(a)). Now, we need to express this coordinate in the form of cos and sin functions to obtain the identity for opposite cosine (-a).

Using the symmetry property of cosine function, we know that the cosine function is an even function. This means that cos(-a) = cos(a). Similarly, the sine function is an odd function, so sin(-a) = -sin(a).

Now, substituting these values into the coordinate expression, we have (-cos(a), -sin(a)) = (-cos(-a), sin(-a)).

Since the point on the unit circle is the representation of cos and sin functions, we can write the identity for opposite cosine (-a) as:

cos(-a) = -cos(a)

This identity states that the cosine of the negative angle is equal to the negative of the cosine of the positive angle.

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