Question

How do you find period and amplitude of $\cos x$ ?

Answer 1

The given question has a single trigonometric function which is multiplied by a constant 2. Nothing has changed in its argument. Now, we know that the amplitude of $\cos x$ is $( - 1,1)$. Since we are multiplying the whole $\cos x$ by 2, our amplitude will also be multiplied by 2. Also, the period of $\cos x$ is $2\pi$. But in order to change the value of period, there should have been a constant multiplied in front of $x$. But since $\cos x$ is $\cos x$ and not $\cos nx$, the period of $y = 2\cos x$ remains the same, that is $2\pi$.

Complete step by step answer:
In order to solve the question, the first thing we need to do is to understand the meaning of amplitude and period.
Amplitude: This is the range of values between which a particular function varies. The range is starting from the lowest possible value and ends at the maximum possible value. So, when you draw the graph of the function, you will see that the values are forever inside the range of max and min only. It will never cross the boundary.

Period: This is the range of the function after which the value starts repeating itself the way it appeared the first time. Suppose, values of a function at different points are 1,2,3,4,5,1,2,3,4,5,1,2,3,4,5 and so on. Clearly we can say that the size of period here is 4 since after every 4 values the queue of numbers starts again from the beginning.Now, the given equation is $y = 2\cos x$.

We know that the amplitude of $\cos x$ is $( - 1,1)$. Since we are multiplying the whole $\cos x$ by 2, our amplitude will also be multiplied by 2.Therefore, the amplitude of $y = 2\cos x$ is equal to$( - 2,2)$. Also, the period of $\cos x$ is $2\pi$. But in order to change the value of period, there should have been a constant multiplied in front of $x$. But since $\cos x$ is $\cos x$and not$\cos nx$, the period of $y = 2\cos x$ remains the same, that is $2\pi$.

Note: Amplitude of a trigonometric function changes whenever you multiply the whole trigonometric function by a constant n. In that case the amplitude also gets multiplied by $n$. And the period of a trigonometric function changes only when you multiply the argument of the trigonometric function by n. In that case, the period also gets multiplied by $n$.