# Given angle x, where 0 <= x < 360 (degrees), cos(x) is equal to a unique value.

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## cos(x) assumes all values between -1 and 1. Each of those values appears twice in a period of the curve. Since you want a unique value,

x = 0 so cos(x) = 1
or
x = 180, so cos(x) = -1

for any other value where cos(x) = k,
cos(360-x) is also equal to k.

Look at the graph and you will see what I mean.

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## Sure, let's solve this problem together. Before we start, I must warn you that math and I have a love-hate relationship. So, brace yourself for some clownish reasoning.

The cosine function is periodic with a period of 360 degrees. That means that if we find one angle x where cos(x) has a certain value, we can find an infinite number of other angles that also have the same cosine value.

Let's take a look at the unit circle, shall we? Imagine a clown spinning around on a unicycle. Now, picture a point on the edge of the circle representing angle x. The x-coordinate of that point is equal to cos(x). Hmm... I can smell the circus peanuts already!

So, let's say that cos(x) equals some unique value, let's call it "c". Now, here's where things get interesting. The cosine function is positive in the first and fourth quadrants of the unit circle, and negative in the second and third quadrants. So, if cos(x) = c, we can conclude that x could be in the first, second, third, or fourth quadrant, which means we have four possible solutions.

To obtain the approximate value of x, we can use a little trigonometric trickery. We know that the cosine function is symmetrical around the y-axis, which means that cos(x) = cos(-x). That allows us to find two angles in the first and fourth quadrants whose cosine values are both "c". We'll call them x1 and x2.

Now, to find the values of x1 and x2, we can use the inverse cosine function, also known as arc cosine or cos^(-1). This function gives us the angle whose cosine is a given value. In mathematical notation, it looks something like this: x = cos^(-1)(c).

In our case, we're looking for the nearest degree, so we round x1 and x2 to the nearest degree to get two approximate solutions. Then, we can claim with confidence (or at least clown-level confidence) that these two values represent the angle x where cos(x) is equal to "c".

I hope this convoluted circus act of an explanation somehow made sense to you. Remember, laughter is the best cosine!

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## To determine the value of cos(x) to the nearest degree, you can use the inverse cosine function, also known as the arccosine function. The arccosine function calculates the angle whose cosine is equal to a given value.

In this case, since you are given the value of cos(x), you can use the arccosine function to find the value of x.

Using a scientific calculator or a mathematical software, calculate the arccosine of the given value of cos(x). This will give you the angle whose cosine is equal to the given value.

For example, if cos(x) = 0.5, you would calculate arccos(0.5), which yields 60 degrees. Therefore, the value of x to the nearest degree would be 60 degrees.

Justification:
The cosine function is periodic with a period of 360 degrees. This means that the cosine of an angle and its corresponding angle shifted by 360 degrees have the same value.

For example, cos(60 degrees) = cos(60 + 360 degrees) = 0.5

However, within the range 0 <= x < 360 degrees, the cosine function has a unique value. As you move along the unit circle in the counterclockwise direction starting from the positive x-axis, the cosine function starts at 1, decreases to -1 at 180 degrees, and then increases back to 1 at 360 degrees. Within this range, the cosine function takes every value between -1 and 1 exactly once.

Therefore, by using the arccosine function, we can uniquely determine the value of x for a given value of cos(x) within the range 0 <= x < 360 degrees.