Question

How do you find the amplitude of a cosine function?

Answer 1

Hint:
 We start solving the problem by recalling the definition of the amplitude of the cosine or sine function as half the distance between the maximum and minimum values of the function (or) as the distance the center line to the maximum or minimum of the function. We then consider the cosine function $ y=a\cos \left( bx \right) $ and find half the distance between maximum and minimum values of it based on the range of the function. We then multiply the obtained distance with the modulus of a to get the required answer.

Complete step by step answer:
According to the problem, we are asked to find the amplitude of a cosine function.
Let us recall the definition of the amplitude of a sine or a cosine function.
We know that the amplitude of the cosine or sine function is defined as half the distance between the maximum and minimum values of the function. (or)
Amplitude of a sine or cosine function is defined as the distance the center line to the maximum or minimum of the function.
So, the amplitude of the function \[y=a\sin \left( bx \right)\] or $ y=a\cos \left( bx \right) $ is defined as $ \left| a \right| $ , where $ \left| . \right| $ represents modulus function. Since the range of $ \sin \left( bx \right) $ or $ \cos \left( bx \right) $ is $ \left[ -1,1 \right] $ and half the distance between maximum and minimum values is 1.
 $ \therefore $ We have found that the amplitude of a cosine function $ y=a\cos \left( bx \right) $ as $ \left| a \right| $ .

Note:
We can also solve this problem by drawing the plot of the cosine function which will also give a similar solution following the definition. Whenever we get this type of problem, we first try to find half the distance between the maximum and minimum values of the function. We should know that the amplitude of any function is positive and considering negative is not correct. Similarly, we can expect problems to find the period of a sine or cosine function.