Question

How do I find the value of $ \cos \left( {\dfrac{{3\pi }}{2}} \right) $ ?

Answer 1

Hint: Here we need to know about the graph of $ \cos x $ and from that we will get to know that value of $ \cos \left( {\dfrac{{(2n + 1)\pi }}{2}} \right) = 0,n \in Z $ and therefore we can say that for every integer belonging to $ n $ $ \cos \left( {\dfrac{{(2n + 1)\pi }}{2}} \right) = 0 $

Complete step-by-step answer:
Here we are given to find the value of $ \cos \left( {\dfrac{{3\pi }}{2}} \right) $
So we must know that the graph of $ \cos x $ look like as given below:

Here we can observe that for value of 
 $
  \cos 0 = 1 \\
  \cos 90 = \cos \dfrac{\pi }{2} = 0 \\ 
  $
So we can also notice that on the $ x - {\text{axis}} $ , we have the angles whose trigonometric function $ \cos $ is to be taken and on the $ y - {\text{axis}} $ , we have the value of the trigonometric function after taking the function $ \cos $ of that value. Hence we can say that from the graph that the value of integral odd multiples of $ \dfrac{\pi }{2} $ is zero as per the graph of the trigonometric function.
Hence we can say that value of $ \cos \left( {\dfrac{{(2n + 1)\pi }}{2}} \right) = 0,n \in Z $
So when we put $ n = 0 $ we get
 $ \cos \left( {\dfrac{{(2n + 1)\pi }}{2}} \right) = \cos \dfrac{\pi }{2} = 0 $
Now we can put $ n = 1 $ we will get:
 $ \cos \left( {\dfrac{{(2n + 1)\pi }}{2}} \right) = \cos \dfrac{{3\pi }}{2} = 0 $
Hence we have got the value of the required function which is given in the problem as $ \cos \dfrac{{3\pi }}{2} = 0 $
So we need to know that whenever we have the odd multiple of $ \dfrac{\pi }{2} $ and we need to find the cosine of that function then directly we can write the value of that trigonometric function to be zero. Hence we must keep in mind all the graphs of the trigonometric functions as they make the solution very simple.

Note: Here the student must have the knowledge of all the graphs of the trigonometric functions so that the value of that function can be calculated easily. We can also do it in other way by writing the function as:
 $ \cos \dfrac{{3\pi }}{2} = \cos \left( {\pi + \dfrac{\pi }{2}} \right) $
Now we will apply the trigonometric property $ \cos (\pi + \theta ) = - \cos \theta $
So we will get $ \cos \dfrac{{3\pi }}{2} = \cos \left( {\pi + \dfrac{\pi }{2}} \right) = - \cos \dfrac{\pi }{2} = 0 $