Note that the cosine function cos(x) is a trigonometric function. Here, we will learn about the domain, range, and period of cos(x).
Domain of cos x
One knows that for any value of x, the cosine function cos(x) is defined. Thus, by the definition of the domain of a function, we can say that the domain of cos(x) is the set of all real numbers. In other words,
The domain of cos(x) = $(-\infty, \infty)$
For any real number $a$, the function cos(ax) is defined for all real numbers. So the domain of cos(ax) is the set of all real numbers. For example,
- Domain of cos(2x) = $(-\infty, \infty)$
- Domain of cos(3x) = $(-\infty, \infty)$
- Domain of cos(4x) = $(-\infty, \infty)$
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Range of cos x
From the graph y=cos(x), one can see that the value of cos(x) oscillates between -1 and 1. That is, we have $-1 \leq \cos x \leq 1$ for all real numbers x. Thus, the range of cos(x) is the closed interval [-1, 1].
Range of cos(x) = [-1, 1]
The range of cos(ax) is the closed interval [-1, 1]. For example,
- Range of cos(2x) = [-1, 1]
- Range of cos(3x) = [-1, 1]
- Range of cos(4x) = [-1, 1]
Period of cos x
For any natural number n, it is known that
$\cos(2n\pi+x)=\cos x$.
From this, we see that $2\pi$ is the smallest number (corresponding to n=1) such that
$\cos(2\pi+x)=\cos x$.
Thus, $2\pi$ is the period of cos(x).
Question 1: Find the period of cos(ax).
Solution:
The period of cos(ax) is $\dfrac{2\pi}{|a|}$. For example, the period of cos(3x) is equal to $\dfrac{2\pi}{3}$.
Question 2: Find the period of cos(8x)-cos(2x).
Solution:
The period of cos(8x) is $\dfrac{2\pi}{8}$ $=\frac{\pi}{4}$ and the period of cos(2x) is $\dfrac{2\pi}{2}=\pi$.
As the period of the difference of two periodic functions is the least common multiple (LCM) of their periods, we conclude that the period of cos(8x)-cos(2x) is
= LCM $\{\dfrac{\pi}{4}, \pi \}$
= $\pi$.
Therefore, the period of cos(8x)-cos(2x) is $\pi$.
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