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The cosine function is one of the fundamental trigonometric functions, frequently used to determine relationships in geometric contexts. When we're working with angles, particularly in a circle, the cosine of an angle provides the horizontal coordinate of a point on the unit circle. The unit circle is simply a circle with a radius of one, centered at the origin of a coordinate system.
When considering the cosine function, it's important to be aware of its periodic nature. This means that its values repeat at regular intervals, specifically every full cycle of the circle, or every 360° (which is equivalent to 2π radians).
In terms of functionality, cosine is an **even function**. This property plays a crucial role in determining the cosine values of negative angles, such as in our exercise on the angle of \(-\frac{3\pi}{4}\). Understanding how the cosine function works, both in terms of its values and its qualities as an even function, allows us to simplify the task of finding trigonometric values.

In mathematics, functions can be classified as either even, odd, or neither, based on their symmetry properties. The cosine function is an example of an **even function**. This means that its graph is symmetric with respect to the y-axis. Mathematically, this is represented by the equation \( \cos(-x) = \cos(x) \). This is because, for every point (x, y) on the graph, there is a point (-x, y).
Odd functions, on the other hand, are symmetric around the origin. One common odd function is the sine, with the property \( \sin(-x) = -\sin(x) \). These concepts play a fundamental role when resolving trigonometric problems involving negative angle measures, as they simplify the computations allowing us to find values efficiently.
Recognizing whether a function is even or odd helps streamline your process of working with negative angles, reducing potential errors in determination.

The reference angle is another pivotal concept when working with trigonometry. It is defined as the smallest angle that a given angle makes with the x-axis. This angle is always positive and less than or equal to \( \frac{\pi}{2} \) radians (or 90°). Reference angles are an important tool because they allow us to find the trigonometric values of angles outside the first quadrant.
For example, when dealing with an angle like \(-\frac{3\pi}{4}\), we identify its reference angle by taking its positive counterpart \(\frac{3\pi}{4}\). This angle lies in the second quadrant, where cosine values are negative. By understanding where these angles lie on the unit circle, we can use reference angles to find the exact value of trigonometric functions more easily.
By combining your understanding of the unit circle, reference angles, and the nature of trigonometric functions, you solidify your comprehension of trigonometric values for any angle.